Calculus Examples

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

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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 $g (x) = {(x - 2)}^{3}$ .

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

Step 1.1.1.3

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 $g (x) = x - 2$ .

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Step 1.1.1.3.1

To apply the Chain Rule, set $u$ as $x - 2$ .

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

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

Replace all occurrences of $u$ with $x - 2$ .

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

Step 1.1.1.4

Differentiate.

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Step 1.1.1.4.1

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

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

Step 1.1.1.4.2

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

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

Step 1.1.1.4.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 1.1.1.4.4

Simplify the expression.

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Step 1.1.1.4.4.1

Add $1$ and $0$ .

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

Step 1.1.1.4.4.2

Multiply $3$ by $1$ .

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

Step 1.1.1.4.5

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

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

Step 1.1.1.4.6

Multiply ${(x - 2)}^{3}$ by $1$ .

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

Step 1.1.1.5

Simplify.

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Step 1.1.1.5.1

Apply the distributive property.

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

Step 1.1.1.5.2

Multiply $3$ by $3$ .

$f' (x) = 9 (x ({(x - 2)}^{2})) + 3 {(x - 2)}^{3}$

Step 1.1.1.5.3

Factor $3 {(x - 2)}^{2}$ out of $9 x {(x - 2)}^{2} + 3 {(x - 2)}^{3}$ .

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Step 1.1.1.5.3.1

Factor $3 {(x - 2)}^{2}$ out of $9 x {(x - 2)}^{2}$ .

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

Step 1.1.1.5.3.2

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

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

Step 1.1.1.5.3.3

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

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

Step 1.1.1.5.4

Add $3 x$ and $x$ .

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

Step 1.1.1.5.5

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

$f' (x) = 3 ((x - 2) (x - 2)) (4 x - 2)$

Step 1.1.1.5.6

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 1.1.1.5.6.1

Apply the distributive property.

$f' (x) = 3 (x (x - 2) - 2 (x - 2)) (4 x - 2)$

Step 1.1.1.5.6.2

Apply the distributive property.

$f' (x) = 3 (x \cdot x + x \cdot - 2 - 2 (x - 2)) (4 x - 2)$

Step 1.1.1.5.6.3

Apply the distributive property.

$f' (x) = 3 (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) (4 x - 2)$

Step 1.1.1.5.7

Simplify and combine like terms.

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Step 1.1.1.5.7.1

Simplify each term.

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Step 1.1.1.5.7.1.1

Multiply $x$ by $x$ .

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

Step 1.1.1.5.7.1.2

Move $- 2$ to the left of $x$ .

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

Step 1.1.1.5.7.1.3

Multiply $- 2$ by $- 2$ .

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

Step 1.1.1.5.7.2

Subtract $2 x$ from $- 2 x$ .

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

Step 1.1.1.5.8

Apply the distributive property.

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

Step 1.1.1.5.9

Simplify.

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Step 1.1.1.5.9.1

Multiply $- 4$ by $3$ .

$f' (x) = (3 x^{2} - 12 x + 3 \cdot 4) (4 x - 2)$

Step 1.1.1.5.9.2

Multiply $3$ by $4$ .

$f' (x) = (3 x^{2} - 12 x + 12) (4 x - 2)$

Step 1.1.1.5.10

Expand $(3 x^{2} - 12 x + 12) (4 x - 2)$ by multiplying each term in the first expression by each term in the second expression.

$f' (x) = 3 x^{2} (4 x) + 3 x^{2} \cdot - 2 - 12 x (4 x) - 12 x \cdot - 2 + 12 (4 x) + 12 \cdot - 2$

Step 1.1.1.5.11

Simplify each term.

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Step 1.1.1.5.11.1

Rewrite using the commutative property of multiplication.

$f' (x) = 3 \cdot (4 x^{2} x) + 3 x^{2} \cdot - 2 - 12 x (4 x) - 12 x \cdot - 2 + 12 (4 x) + 12 \cdot - 2$

Step 1.1.1.5.11.2

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

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Step 1.1.1.5.11.2.1

Move $x$ .

$f' (x) = 3 \cdot (4 (x \cdot x^{2})) + 3 x^{2} \cdot - 2 - 12 x (4 x) - 12 x \cdot - 2 + 12 (4 x) + 12 \cdot - 2$

Step 1.1.1.5.11.2.2

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

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Step 1.1.1.5.11.2.2.1

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

$f' (x) = 3 \cdot (4 (x \cdot x^{2})) + 3 x^{2} \cdot - 2 - 12 x (4 x) - 12 x \cdot - 2 + 12 (4 x) + 12 \cdot - 2$

Step 1.1.1.5.11.2.2.2

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

$f' (x) = 3 \cdot (4 x^{1 + 2}) + 3 x^{2} \cdot - 2 - 12 x (4 x) - 12 x \cdot - 2 + 12 (4 x) + 12 \cdot - 2$

Step 1.1.1.5.11.2.3

Add $1$ and $2$ .

$f' (x) = 3 \cdot (4 x^{3}) + 3 x^{2} \cdot - 2 - 12 x (4 x) - 12 x \cdot - 2 + 12 (4 x) + 12 \cdot - 2$

Step 1.1.1.5.11.3

Multiply $3$ by $4$ .

$f' (x) = 12 x^{3} + 3 x^{2} \cdot - 2 - 12 x (4 x) - 12 x \cdot - 2 + 12 (4 x) + 12 \cdot - 2$

Step 1.1.1.5.11.4

Multiply $- 2$ by $3$ .

$f' (x) = 12 x^{3} - 6 x^{2} - 12 x (4 x) - 12 x \cdot - 2 + 12 (4 x) + 12 \cdot - 2$

Step 1.1.1.5.11.5

Rewrite using the commutative property of multiplication.

$f' (x) = 12 x^{3} - 6 x^{2} - 12 \cdot (4 x \cdot x) - 12 x \cdot - 2 + 12 (4 x) + 12 \cdot - 2$

Step 1.1.1.5.11.6

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

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Step 1.1.1.5.11.6.1

Move $x$ .

$f' (x) = 12 x^{3} - 6 x^{2} - 12 \cdot (4 (x \cdot x)) - 12 x \cdot - 2 + 12 (4 x) + 12 \cdot - 2$

Step 1.1.1.5.11.6.2

Multiply $x$ by $x$ .

$f' (x) = 12 x^{3} - 6 x^{2} - 12 \cdot (4 x^{2}) - 12 x \cdot - 2 + 12 (4 x) + 12 \cdot - 2$

Step 1.1.1.5.11.7

Multiply $- 12$ by $4$ .

$f' (x) = 12 x^{3} - 6 x^{2} - 48 x^{2} - 12 x \cdot - 2 + 12 (4 x) + 12 \cdot - 2$

Step 1.1.1.5.11.8

Multiply $- 2$ by $- 12$ .

$f' (x) = 12 x^{3} - 6 x^{2} - 48 x^{2} + 24 x + 12 (4 x) + 12 \cdot - 2$

Step 1.1.1.5.11.9

Multiply $4$ by $12$ .

$f' (x) = 12 x^{3} - 6 x^{2} - 48 x^{2} + 24 x + 48 x + 12 \cdot - 2$

Step 1.1.1.5.11.10

Multiply $12$ by $- 2$ .

$f' (x) = 12 x^{3} - 6 x^{2} - 48 x^{2} + 24 x + 48 x - 24$

Step 1.1.1.5.12

Subtract $48 x^{2}$ from $- 6 x^{2}$ .

$f' (x) = 12 x^{3} - 54 x^{2} + 24 x + 48 x - 24$

Step 1.1.1.5.13

Add $24 x$ and $48 x$ .

$f' (x) = 12 x^{3} - 54 x^{2} + 72 x - 24$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $12 x^{3} - 54 x^{2} + 72 x - 24$ with respect to $x$ is $\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [- 54 x^{2}] + \frac{d}{d x} [72 x] + \frac{d}{d x} [- 24]$ .

$f'' (x) = \frac{d}{d x} (12 x^{3}) + \frac{d}{d x} (- 54 x^{2}) + \frac{d}{d x} (72 x) + \frac{d}{d x} (- 24)$

Step 1.1.2.2

Evaluate $\frac{d}{d x} [12 x^{3}]$ .

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Step 1.1.2.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{3}$ with respect to $x$ is $12 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 12 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 54 x^{2}) + \frac{d}{d x} (72 x) + \frac{d}{d x} (- 24)$

Step 1.1.2.2.2

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

$f'' (x) = 12 (3 x^{2}) + \frac{d}{d x} (- 54 x^{2}) + \frac{d}{d x} (72 x) + \frac{d}{d x} (- 24)$

Step 1.1.2.2.3

Multiply $3$ by $12$ .

$f'' (x) = 36 x^{2} + \frac{d}{d x} (- 54 x^{2}) + \frac{d}{d x} (72 x) + \frac{d}{d x} (- 24)$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [- 54 x^{2}]$ .

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Step 1.1.2.3.1

Since $- 54$ is constant with respect to $x$ , the derivative of $- 54 x^{2}$ with respect to $x$ is $- 54 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 36 x^{2} - 54 \frac{d}{d x} x^{2} + \frac{d}{d x} (72 x) + \frac{d}{d x} (- 24)$

Step 1.1.2.3.2

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

$f'' (x) = 36 x^{2} - 54 (2 x) + \frac{d}{d x} (72 x) + \frac{d}{d x} (- 24)$

Step 1.1.2.3.3

Multiply $2$ by $- 54$ .

$f'' (x) = 36 x^{2} - 108 x + \frac{d}{d x} (72 x) + \frac{d}{d x} (- 24)$

Step 1.1.2.4

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

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Step 1.1.2.4.1

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

$f'' (x) = 36 x^{2} - 108 x + 72 \frac{d}{d x} (x) + \frac{d}{d x} (- 24)$

Step 1.1.2.4.2

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

$f'' (x) = 36 x^{2} - 108 x + 72 \cdot 1 + \frac{d}{d x} (- 24)$

Step 1.1.2.4.3

Multiply $72$ by $1$ .

$f'' (x) = 36 x^{2} - 108 x + 72 + \frac{d}{d x} (- 24)$

Step 1.1.2.5

Differentiate using the Constant Rule.

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Step 1.1.2.5.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24$ with respect to $x$ is $0$ .

$f'' (x) = 36 x^{2} - 108 x + 72 + 0$

Step 1.1.2.5.2

Add $36 x^{2} - 108 x + 72$ and $0$ .

$f'' (x) = 36 x^{2} - 108 x + 72$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $36 x^{2} - 108 x + 72$ .

$36 x^{2} - 108 x + 72$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $36 x^{2} - 108 x + 72 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$36 x^{2} - 108 x + 72 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $36$ out of $36 x^{2} - 108 x + 72$ .

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Step 1.2.2.1.1

Factor $36$ out of $36 x^{2}$ .

$36 (x^{2}) - 108 x + 72 = 0$

Step 1.2.2.1.2

Factor $36$ out of $- 108 x$ .

$36 (x^{2}) + 36 (- 3 x) + 72 = 0$

Step 1.2.2.1.3

Factor $36$ out of $72$ .

$36 x^{2} + 36 (- 3 x) + 36 \cdot 2 = 0$

Step 1.2.2.1.4

Factor $36$ out of $36 x^{2} + 36 (- 3 x)$ .

$36 (x^{2} - 3 x) + 36 \cdot 2 = 0$

Step 1.2.2.1.5

Factor $36$ out of $36 (x^{2} - 3 x) + 36 \cdot 2$ .

$36 (x^{2} - 3 x + 2) = 0$

Step 1.2.2.2

Factor.

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Step 1.2.2.2.1

Factor $x^{2} - 3 x + 2$ using the AC method.

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Step 1.2.2.2.1.1

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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 1.2.2.2.1.2

Write the factored form using these integers.

$36 ((x - 2) (x - 1)) = 0$

Step 1.2.2.2.2

Remove unnecessary parentheses.

$36 (x - 2) (x - 1) = 0$

Step 1.2.3

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

$x - 2 = 0$

$x - 1 = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$x - 2 = 0$

Step 1.2.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.2.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.6

The final solution is all the values that make $36 (x - 2) (x - 1) = 0$ true.

$x = 2, 1$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, 1) \cup (1, 2) \cup (2, \infty)$

Step 4

Substitute any number from the interval $(- \infty, 1)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

Replace the variable $x$ with $0$ in the expression.

$f'' (0) = 36 {(0)}^{2} - 108 \cdot 0 + 72$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$f'' (0) = 36 \cdot 0 - 108 \cdot 0 + 72$

Step 4.2.1.2

Multiply $36$ by $0$ .

$f'' (0) = 0 - 108 \cdot 0 + 72$

Step 4.2.1.3

Multiply $- 108$ by $0$ .

$f'' (0) = 0 + 0 + 72$

Step 4.2.2

Simplify by adding numbers.

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Step 4.2.2.1

Add $0$ and $0$ .

$f'' (0) = 0 + 72$

Step 4.2.2.2

Add $0$ and $72$ .

$f'' (0) = 72$

Step 4.2.3

The final answer is $72$ .

$72$

Step 4.3

The graph is concave up on the interval $(- \infty, 1)$ because $f'' (0)$ is positive.

Concave up on $(- \infty, 1)$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(1, 2)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $1.5$ in the expression.

$f'' (1.5) = 36 {(1.5)}^{2} - 108 \cdot 1.5 + 72$

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 $1.5$ to the power of $2$ .

$f'' (1.5) = 36 \cdot 2.25 - 108 \cdot 1.5 + 72$

Step 5.2.1.2

Multiply $36$ by $2.25$ .

$f'' (1.5) = 81 - 108 \cdot 1.5 + 72$

Step 5.2.1.3

Multiply $- 108$ by $1.5$ .

$f'' (1.5) = 81 - 162 + 72$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $162$ from $81$ .

$f'' (1.5) = - 81 + 72$

Step 5.2.2.2

Add $- 81$ and $72$ .

$f'' (1.5) = - 9$

Step 5.2.3

The final answer is $- 9$ .

$- 9$

Step 5.3

The graph is concave down on the interval $(1, 2)$ because $f'' (1.5)$ is negative.

Concave down on $(1, 2)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(2, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (4) = 36 {(4)}^{2} - 108 \cdot 4 + 72$

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 $4$ to the power of $2$ .

$f'' (4) = 36 \cdot 16 - 108 \cdot 4 + 72$

Step 6.2.1.2

Multiply $36$ by $16$ .

$f'' (4) = 576 - 108 \cdot 4 + 72$

Step 6.2.1.3

Multiply $- 108$ by $4$ .

$f'' (4) = 576 - 432 + 72$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $432$ from $576$ .

$f'' (4) = 144 + 72$

Step 6.2.2.2

Add $144$ and $72$ .

$f'' (4) = 216$

Step 6.2.3

The final answer is $216$ .

$216$

Step 6.3

The graph is concave up on the interval $(2, \infty)$ because $f'' (4)$ is positive.

Concave up on $(2, \infty)$ since $f'' (x)$ is positive

Step 7

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, 1)$ since $f'' (x)$ is positive

Concave down on $(1, 2)$ since $f'' (x)$ is negative

Concave up on $(2, \infty)$ since $f'' (x)$ is positive

Step 8