Calculus Examples

$f (x) = 2 x {(x - 4)}^{3}$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x {(x - 4)}^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x {(x - 4)}^{3}]$ .

$f' (x) = 2 \frac{d}{d x} (x {(x - 4)}^{3})$

Step 1.1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(x - 4)}^{3}$ .

$f' (x) = 2 (x \frac{d}{d x} ({(x - 4)}^{3}) + {(x - 4)}^{3} \frac{d}{d x} (x))$

Step 1.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x - 4$ .

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Step 1.1.3.1

To apply the Chain Rule, set $u$ as $x - 4$ .

$f' (x) = 2 (x (\frac{d}{d u} (u^{3}) \frac{d}{d x} (x - 4)) + {(x - 4)}^{3} \frac{d}{d x} (x))$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$f' (x) = 2 (x (3 u^{2} \frac{d}{d x} (x - 4)) + {(x - 4)}^{3} \frac{d}{d x} (x))$

Step 1.1.3.3

Replace all occurrences of $u$ with $x - 4$ .

$f' (x) = 2 (x (3 {(x - 4)}^{2} \frac{d}{d x} (x - 4)) + {(x - 4)}^{3} \frac{d}{d x} (x))$

Step 1.1.4

Differentiate.

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Step 1.1.4.1

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$f' (x) = 2 (x (3 {(x - 4)}^{2} (\frac{d}{d x} (x) + \frac{d}{d x} (- 4))) + {(x - 4)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = 2 (x (3 {(x - 4)}^{2} (1 + \frac{d}{d x} (- 4))) + {(x - 4)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$f' (x) = 2 (x (3 {(x - 4)}^{2} (1 + 0)) + {(x - 4)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.4

Simplify the expression.

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Step 1.1.4.4.1

Add $1$ and $0$ .

$f' (x) = 2 (x (3 {(x - 4)}^{2} \cdot 1) + {(x - 4)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.4.2

Multiply $3$ by $1$ .

$f' (x) = 2 (x (3 {(x - 4)}^{2}) + {(x - 4)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = 2 (x (3 {(x - 4)}^{2}) + {(x - 4)}^{3} \cdot 1)$

Step 1.1.4.6

Multiply ${(x - 4)}^{3}$ by $1$ .

$f' (x) = 2 (x (3 {(x - 4)}^{2}) + {(x - 4)}^{3})$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Apply the distributive property.

$f' (x) = 2 (x (3 {(x - 4)}^{2})) + 2 {(x - 4)}^{3}$

Step 1.1.5.2

Multiply $3$ by $2$ .

$f' (x) = 6 (x ({(x - 4)}^{2})) + 2 {(x - 4)}^{3}$

Step 1.1.5.3

Factor $2 {(x - 4)}^{2}$ out of $6 x {(x - 4)}^{2} + 2 {(x - 4)}^{3}$ .

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Step 1.1.5.3.1

Factor $2 {(x - 4)}^{2}$ out of $6 x {(x - 4)}^{2}$ .

$f' (x) = 2 {(x - 4)}^{2} (3 x) + 2 {(x - 4)}^{3}$

Step 1.1.5.3.2

Factor $2 {(x - 4)}^{2}$ out of $2 {(x - 4)}^{3}$ .

$f' (x) = 2 {(x - 4)}^{2} (3 x) + 2 {(x - 4)}^{2} (x - 4)$

Step 1.1.5.3.3

Factor $2 {(x - 4)}^{2}$ out of $2 {(x - 4)}^{2} (3 x) + 2 {(x - 4)}^{2} (x - 4)$ .

$f' (x) = 2 {(x - 4)}^{2} (3 x + x - 4)$

Step 1.1.5.4

Add $3 x$ and $x$ .

$f' (x) = 2 {(x - 4)}^{2} (4 x - 4)$

Step 1.1.5.5

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$f' (x) = 2 ((x - 4) (x - 4)) (4 x - 4)$

Step 1.1.5.6

Expand $(x - 4) (x - 4)$ using the FOIL Method.

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Step 1.1.5.6.1

Apply the distributive property.

$f' (x) = 2 (x (x - 4) - 4 (x - 4)) (4 x - 4)$

Step 1.1.5.6.2

Apply the distributive property.

$f' (x) = 2 (x \cdot x + x \cdot - 4 - 4 (x - 4)) (4 x - 4)$

Step 1.1.5.6.3

Apply the distributive property.

$f' (x) = 2 (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) (4 x - 4)$

Step 1.1.5.7

Simplify and combine like terms.

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Step 1.1.5.7.1

Simplify each term.

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Step 1.1.5.7.1.1

Multiply $x$ by $x$ .

$f' (x) = 2 (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4) (4 x - 4)$

Step 1.1.5.7.1.2

Move $- 4$ to the left of $x$ .

$f' (x) = 2 (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4) (4 x - 4)$

Step 1.1.5.7.1.3

Multiply $- 4$ by $- 4$ .

$f' (x) = 2 (x^{2} - 4 x - 4 x + 16) (4 x - 4)$

Step 1.1.5.7.2

Subtract $4 x$ from $- 4 x$ .

$f' (x) = 2 (x^{2} - 8 x + 16) (4 x - 4)$

Step 1.1.5.8

Apply the distributive property.

$f' (x) = (2 x^{2} + 2 (- 8 x) + 2 \cdot 16) (4 x - 4)$

Step 1.1.5.9

Simplify.

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Step 1.1.5.9.1

Multiply $- 8$ by $2$ .

$f' (x) = (2 x^{2} - 16 x + 2 \cdot 16) (4 x - 4)$

Step 1.1.5.9.2

Multiply $2$ by $16$ .

$f' (x) = (2 x^{2} - 16 x + 32) (4 x - 4)$

Step 1.1.5.10

Expand $(2 x^{2} - 16 x + 32) (4 x - 4)$ by multiplying each term in the first expression by each term in the second expression.

$f' (x) = 2 x^{2} (4 x) + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 1.1.5.11

Simplify each term.

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Step 1.1.5.11.1

Rewrite using the commutative property of multiplication.

$f' (x) = 2 \cdot (4 x^{2} x) + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 1.1.5.11.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.5.11.2.1

Move $x$ .

$f' (x) = 2 \cdot (4 (x \cdot x^{2})) + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 1.1.5.11.2.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.5.11.2.2.1

Raise $x$ to the power of $1$ .

$f' (x) = 2 \cdot (4 (x \cdot x^{2})) + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 1.1.5.11.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = 2 \cdot (4 x^{1 + 2}) + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 1.1.5.11.2.3

Add $1$ and $2$ .

$f' (x) = 2 \cdot (4 x^{3}) + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 1.1.5.11.3

Multiply $2$ by $4$ .

$f' (x) = 8 x^{3} + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 1.1.5.11.4

Multiply $- 4$ by $2$ .

$f' (x) = 8 x^{3} - 8 x^{2} - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 1.1.5.11.5

Rewrite using the commutative property of multiplication.

$f' (x) = 8 x^{3} - 8 x^{2} - 16 \cdot (4 x \cdot x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 1.1.5.11.6

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.5.11.6.1

Move $x$ .

$f' (x) = 8 x^{3} - 8 x^{2} - 16 \cdot (4 (x \cdot x)) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 1.1.5.11.6.2

Multiply $x$ by $x$ .

$f' (x) = 8 x^{3} - 8 x^{2} - 16 \cdot (4 x^{2}) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 1.1.5.11.7

Multiply $- 16$ by $4$ .

$f' (x) = 8 x^{3} - 8 x^{2} - 64 x^{2} - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 1.1.5.11.8

Multiply $- 4$ by $- 16$ .

$f' (x) = 8 x^{3} - 8 x^{2} - 64 x^{2} + 64 x + 32 (4 x) + 32 \cdot - 4$

Step 1.1.5.11.9

Multiply $4$ by $32$ .

$f' (x) = 8 x^{3} - 8 x^{2} - 64 x^{2} + 64 x + 128 x + 32 \cdot - 4$

Step 1.1.5.11.10

Multiply $32$ by $- 4$ .

$f' (x) = 8 x^{3} - 8 x^{2} - 64 x^{2} + 64 x + 128 x - 128$

Step 1.1.5.12

Subtract $64 x^{2}$ from $- 8 x^{2}$ .

$f' (x) = 8 x^{3} - 72 x^{2} + 64 x + 128 x - 128$

Step 1.1.5.13

Add $64 x$ and $128 x$ .

$f' (x) = 8 x^{3} - 72 x^{2} + 192 x - 128$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $8 x^{3} - 72 x^{2} + 192 x - 128$ with respect to $x$ is $\frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [- 72 x^{2}] + \frac{d}{d x} [192 x] + \frac{d}{d x} [- 128]$ .

$f'' (x) = \frac{d}{d x} (8 x^{3}) + \frac{d}{d x} (- 72 x^{2}) + \frac{d}{d x} (192 x) + \frac{d}{d x} (- 128)$

Step 1.2.2

Evaluate $\frac{d}{d x} [8 x^{3}]$ .

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Step 1.2.2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{3}$ with respect to $x$ is $8 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 8 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 72 x^{2}) + \frac{d}{d x} (192 x) + \frac{d}{d x} (- 128)$

Step 1.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = 8 (3 x^{2}) + \frac{d}{d x} (- 72 x^{2}) + \frac{d}{d x} (192 x) + \frac{d}{d x} (- 128)$

Step 1.2.2.3

Multiply $3$ by $8$ .

$f'' (x) = 24 x^{2} + \frac{d}{d x} (- 72 x^{2}) + \frac{d}{d x} (192 x) + \frac{d}{d x} (- 128)$

Step 1.2.3

Evaluate $\frac{d}{d x} [- 72 x^{2}]$ .

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Step 1.2.3.1

Since $- 72$ is constant with respect to $x$ , the derivative of $- 72 x^{2}$ with respect to $x$ is $- 72 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 24 x^{2} - 72 \frac{d}{d x} x^{2} + \frac{d}{d x} (192 x) + \frac{d}{d x} (- 128)$

Step 1.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = 24 x^{2} - 72 (2 x) + \frac{d}{d x} (192 x) + \frac{d}{d x} (- 128)$

Step 1.2.3.3

Multiply $2$ by $- 72$ .

$f'' (x) = 24 x^{2} - 144 x + \frac{d}{d x} (192 x) + \frac{d}{d x} (- 128)$

Step 1.2.4

Evaluate $\frac{d}{d x} [192 x]$ .

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Step 1.2.4.1

Since $192$ is constant with respect to $x$ , the derivative of $192 x$ with respect to $x$ is $192 \frac{d}{d x} [x]$ .

$f'' (x) = 24 x^{2} - 144 x + 192 \frac{d}{d x} (x) + \frac{d}{d x} (- 128)$

Step 1.2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = 24 x^{2} - 144 x + 192 \cdot 1 + \frac{d}{d x} (- 128)$

Step 1.2.4.3

Multiply $192$ by $1$ .

$f'' (x) = 24 x^{2} - 144 x + 192 + \frac{d}{d x} (- 128)$

Step 1.2.5

Differentiate using the Constant Rule.

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Step 1.2.5.1

Since $- 128$ is constant with respect to $x$ , the derivative of $- 128$ with respect to $x$ is $0$ .

$f'' (x) = 24 x^{2} - 144 x + 192 + 0$

Step 1.2.5.2

Add $24 x^{2} - 144 x + 192$ and $0$ .

$f'' (x) = 24 x^{2} - 144 x + 192$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $24 x^{2} - 144 x + 192$ .

$24 x^{2} - 144 x + 192$

Step 2

Set the second derivative equal to $0$ then solve the equation $24 x^{2} - 144 x + 192 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$24 x^{2} - 144 x + 192 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $24$ out of $24 x^{2} - 144 x + 192$ .

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Step 2.2.1.1

Factor $24$ out of $24 x^{2}$ .

$24 (x^{2}) - 144 x + 192 = 0$

Step 2.2.1.2

Factor $24$ out of $- 144 x$ .

$24 (x^{2}) + 24 (- 6 x) + 192 = 0$

Step 2.2.1.3

Factor $24$ out of $192$ .

$24 x^{2} + 24 (- 6 x) + 24 \cdot 8 = 0$

Step 2.2.1.4

Factor $24$ out of $24 x^{2} + 24 (- 6 x)$ .

$24 (x^{2} - 6 x) + 24 \cdot 8 = 0$

Step 2.2.1.5

Factor $24$ out of $24 (x^{2} - 6 x) + 24 \cdot 8$ .

$24 (x^{2} - 6 x + 8) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $x^{2} - 6 x + 8$ using the AC method.

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Step 2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 2.2.2.1.2

Write the factored form using these integers.

$24 ((x - 4) (x - 2)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$24 (x - 4) (x - 2) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 4 = 0$

$x - 2 = 0$

Step 2.4

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.6

The final solution is all the values that make $24 (x - 4) (x - 2) = 0$ true.

$x = 4, 2$

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $4$ in $f (x) = 2 x {(x - 4)}^{3}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = 2 (4) {((4) - 4)}^{3}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Multiply $2$ by $4$ .

$f (4) = 8 {((4) - 4)}^{3}$

Step 3.1.2.2

Subtract $4$ from $4$ .

$f (4) = 8 \cdot 0^{3}$

Step 3.1.2.3

Raising $0$ to any positive power yields $0$ .

$f (4) = 8 \cdot 0$

Step 3.1.2.4

Multiply $8$ by $0$ .

$f (4) = 0$

Step 3.1.2.5

The final answer is $0$ .

$0$

Step 3.2

The point found by substituting $4$ in $f (x) = 2 x {(x - 4)}^{3}$ is $(4, 0)$ . This point can be an inflection point.

$(4, 0)$

Step 3.3

Substitute $2$ in $f (x) = 2 x {(x - 4)}^{3}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = 2 (2) {((2) - 4)}^{3}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Multiply $2$ by $2$ .

$f (2) = 4 {((2) - 4)}^{3}$

Step 3.3.2.2

Subtract $4$ from $2$ .

$f (2) = 4 {(- 2)}^{3}$

Step 3.3.2.3

Raise $- 2$ to the power of $3$ .

$f (2) = 4 \cdot - 8$

Step 3.3.2.4

Multiply $4$ by $- 8$ .

$f (2) = - 32$

Step 3.3.2.5

The final answer is $- 32$ .

$- 32$

Step 3.4

The point found by substituting $2$ in $f (x) = 2 x {(x - 4)}^{3}$ is $(2, - 32)$ . This point can be an inflection point.

$(2, - 32)$

Step 3.5

Determine the points that could be inflection points.

$(4, 0), (2, - 32)$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, 2) \cup (2, 4) \cup (4, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 2)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $1.9$ in the expression.

$f'' (1.9) = 24 {(1.9)}^{2} - 144 \cdot 1.9 + 192$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $1.9$ to the power of $2$ .

$f'' (1.9) = 24 \cdot 3.61 - 144 \cdot 1.9 + 192$

Step 5.2.1.2

Multiply $24$ by $3.61$ .

$f'' (1.9) = 86.64 - 144 \cdot 1.9 + 192$

Step 5.2.1.3

Multiply $- 144$ by $1.9$ .

$f'' (1.9) = 86.64 - 273.6 + 192$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $273.6$ from $86.64$ .

$f'' (1.9) = - 186.96 + 192$

Step 5.2.2.2

Add $- 186.96$ and $192$ .

$f'' (1.9) = 5.04$

Step 5.2.3

The final answer is $5.04$ .

$5.04$

Step 5.3

At $1.9$ , the second derivative is $5.04$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 2)$ .

Increasing on $(- \infty, 2)$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(2, 4)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $3$ in the expression.

$f'' (3) = 24 {(3)}^{2} - 144 \cdot 3 + 192$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $3$ to the power of $2$ .

$f'' (3) = 24 \cdot 9 - 144 \cdot 3 + 192$

Step 6.2.1.2

Multiply $24$ by $9$ .

$f'' (3) = 216 - 144 \cdot 3 + 192$

Step 6.2.1.3

Multiply $- 144$ by $3$ .

$f'' (3) = 216 - 432 + 192$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $432$ from $216$ .

$f'' (3) = - 216 + 192$

Step 6.2.2.2

Add $- 216$ and $192$ .

$f'' (3) = - 24$

Step 6.2.3

The final answer is $- 24$ .

$- 24$

Step 6.3

At $3$ , the second derivative is $- 24$ . Since this is negative, the second derivative is decreasing on the interval $(2, 4)$

Decreasing on $(2, 4)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(4, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $4.1$ in the expression.

$f'' (4.1) = 24 {(4.1)}^{2} - 144 \cdot 4.1 + 192$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $4.1$ to the power of $2$ .

$f'' (4.1) = 24 \cdot 16.81 - 144 \cdot 4.1 + 192$

Step 7.2.1.2

Multiply $24$ by $16.81$ .

$f'' (4.1) = 403.44 - 144 \cdot 4.1 + 192$

Step 7.2.1.3

Multiply $- 144$ by $4.1$ .

$f'' (4.1) = 403.44 - 590.4 + 192$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $590.4$ from $403.44$ .

$f'' (4.1) = - 186.96 + 192$

Step 7.2.2.2

Add $- 186.96$ and $192$ .

$f'' (4.1) = 5.04$

Step 7.2.3

The final answer is $5.04$ .

$5.04$

Step 7.3

At $4.1$ , the second derivative is $5.04$ . Since this is positive, the second derivative is increasing on the interval $(4, \infty)$ .

Increasing on $(4, \infty)$ since $f'' (x) > 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(2, - 32), (4, 0)$ .

$(2, - 32), (4, 0)$

Step 9