Calculus Examples

$x^{4} - 8 x^{3} + 18 x^{2}$

Step 1

Write $x^{4} - 8 x^{3} + 18 x^{2}$ as a function.

$f (x) = x^{4} - 8 x^{3} + 18 x^{2}$

Step 2

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

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

Differentiate.

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Step 2.1.1.1.1

By the Sum Rule, the derivative of $x^{4} - 8 x^{3} + 18 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [18 x^{2}]$ .

$f' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 8 x^{3}) + \frac{d}{d x} (18 x^{2})$

Step 2.1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f' (x) = 4 x^{3} + \frac{d}{d x} (- 8 x^{3}) + \frac{d}{d x} (18 x^{2})$

Step 2.1.1.2

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

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Step 2.1.1.2.1

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

$f' (x) = 4 x^{3} - 8 \frac{d}{d x} x^{3} + \frac{d}{d x} (18 x^{2})$

Step 2.1.1.2.2

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

$f' (x) = 4 x^{3} - 8 (3 x^{2}) + \frac{d}{d x} (18 x^{2})$

Step 2.1.1.2.3

Multiply $3$ by $- 8$ .

$f' (x) = 4 x^{3} - 24 x^{2} + \frac{d}{d x} (18 x^{2})$

Step 2.1.1.3

Evaluate $\frac{d}{d x} [18 x^{2}]$ .

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Step 2.1.1.3.1

Since $18$ is constant with respect to $x$ , the derivative of $18 x^{2}$ with respect to $x$ is $18 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 4 x^{3} - 24 x^{2} + 18 \frac{d}{d x} (x^{2})$

Step 2.1.1.3.2

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

$f' (x) = 4 x^{3} - 24 x^{2} + 18 (2 x)$

Step 2.1.1.3.3

Multiply $2$ by $18$ .

$f' (x) = 4 x^{3} - 24 x^{2} + 36 x$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

By the Sum Rule, the derivative of $4 x^{3} - 24 x^{2} + 36 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 24 x^{2}] + \frac{d}{d x} [36 x]$ .

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (36 x)$

Step 2.1.2.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 2.1.2.2.1

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

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (36 x)$

Step 2.1.2.2.2

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

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (36 x)$

Step 2.1.2.2.3

Multiply $3$ by $4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (36 x)$

Step 2.1.2.3

Evaluate $\frac{d}{d x} [- 24 x^{2}]$ .

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Step 2.1.2.3.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x^{2}$ with respect to $x$ is $- 24 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 x^{2} - 24 \frac{d}{d x} x^{2} + \frac{d}{d x} (36 x)$

Step 2.1.2.3.2

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

$f'' (x) = 12 x^{2} - 24 (2 x) + \frac{d}{d x} (36 x)$

Step 2.1.2.3.3

Multiply $2$ by $- 24$ .

$f'' (x) = 12 x^{2} - 48 x + \frac{d}{d x} (36 x)$

Step 2.1.2.4

Evaluate $\frac{d}{d x} [36 x]$ .

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Step 2.1.2.4.1

Since $36$ is constant with respect to $x$ , the derivative of $36 x$ with respect to $x$ is $36 \frac{d}{d x} [x]$ .

$f'' (x) = 12 x^{2} - 48 x + 36 \frac{d}{d x} (x)$

Step 2.1.2.4.2

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

$f'' (x) = 12 x^{2} - 48 x + 36 \cdot 1$

Step 2.1.2.4.3

Multiply $36$ by $1$ .

$f'' (x) = 12 x^{2} - 48 x + 36$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 48 x + 36$ .

$12 x^{2} - 48 x + 36$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $12 x^{2} - 48 x + 36 = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$12 x^{2} - 48 x + 36 = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $12$ out of $12 x^{2} - 48 x + 36$ .

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Step 2.2.2.1.1

Factor $12$ out of $12 x^{2}$ .

$12 (x^{2}) - 48 x + 36 = 0$

Step 2.2.2.1.2

Factor $12$ out of $- 48 x$ .

$12 (x^{2}) + 12 (- 4 x) + 36 = 0$

Step 2.2.2.1.3

Factor $12$ out of $36$ .

$12 x^{2} + 12 (- 4 x) + 12 \cdot 3 = 0$

Step 2.2.2.1.4

Factor $12$ out of $12 x^{2} + 12 (- 4 x)$ .

$12 (x^{2} - 4 x) + 12 \cdot 3 = 0$

Step 2.2.2.1.5

Factor $12$ out of $12 (x^{2} - 4 x) + 12 \cdot 3$ .

$12 (x^{2} - 4 x + 3) = 0$

Step 2.2.2.2

Factor.

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Step 2.2.2.2.1

Factor $x^{2} - 4 x + 3$ using the AC method.

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Step 2.2.2.2.1.1

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 $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 2.2.2.2.1.2

Write the factored form using these integers.

$12 ((x - 3) (x - 1)) = 0$

Step 2.2.2.2.2

Remove unnecessary parentheses.

$12 (x - 3) (x - 1) = 0$

Step 2.2.3

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

$x - 3 = 0$

$x - 1 = 0$

Step 2.2.4

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.2.4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.2.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.2.5

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

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Step 2.2.5.1

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

$x - 1 = 0$

Step 2.2.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.2.6

The final solution is all the values that make $12 (x - 3) (x - 1) = 0$ true.

$x = 3, 1$

Step 3

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 4

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

$(- \infty, 1) \cup (1, 3) \cup (3, \infty)$

Step 5

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

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Step 5.1

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

$f'' (0) = 12 {(0)}^{2} - 48 \cdot 0 + 36$

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

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

$f'' (0) = 12 \cdot 0 - 48 \cdot 0 + 36$

Step 5.2.1.2

Multiply $12$ by $0$ .

$f'' (0) = 0 - 48 \cdot 0 + 36$

Step 5.2.1.3

Multiply $- 48$ by $0$ .

$f'' (0) = 0 + 0 + 36$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add $0$ and $0$ .

$f'' (0) = 0 + 36$

Step 5.2.2.2

Add $0$ and $36$ .

$f'' (0) = 36$

Step 5.2.3

The final answer is $36$ .

$36$

Step 5.3

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 6

Substitute any number from the interval $(1, 3)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (2) = 12 {(2)}^{2} - 48 \cdot 2 + 36$

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 $2$ to the power of $2$ .

$f'' (2) = 12 \cdot 4 - 48 \cdot 2 + 36$

Step 6.2.1.2

Multiply $12$ by $4$ .

$f'' (2) = 48 - 48 \cdot 2 + 36$

Step 6.2.1.3

Multiply $- 48$ by $2$ .

$f'' (2) = 48 - 96 + 36$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $96$ from $48$ .

$f'' (2) = - 48 + 36$

Step 6.2.2.2

Add $- 48$ and $36$ .

$f'' (2) = - 12$

Step 6.2.3

The final answer is $- 12$ .

$- 12$

Step 6.3

The graph is concave down on the interval $(1, 3)$ because $f'' (2)$ is negative.

Concave down on $(1, 3)$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(3, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable $x$ with $6$ in the expression.

$f'' (6) = 12 {(6)}^{2} - 48 \cdot 6 + 36$

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 $6$ to the power of $2$ .

$f'' (6) = 12 \cdot 36 - 48 \cdot 6 + 36$

Step 7.2.1.2

Multiply $12$ by $36$ .

$f'' (6) = 432 - 48 \cdot 6 + 36$

Step 7.2.1.3

Multiply $- 48$ by $6$ .

$f'' (6) = 432 - 288 + 36$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $288$ from $432$ .

$f'' (6) = 144 + 36$

Step 7.2.2.2

Add $144$ and $36$ .

$f'' (6) = 180$

Step 7.2.3

The final answer is $180$ .

$180$

Step 7.3

The graph is concave up on the interval $(3, \infty)$ because $f'' (6)$ is positive.

Concave up on $(3, \infty)$ since $f'' (x)$ is positive

Step 8

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 $(1, 3)$ since $f'' (x)$ is negative

Concave up on $(3, \infty)$ since $f'' (x)$ is positive

Step 9