Calculus Examples

$f (x) = x^{3} - 6 x^{2} + 9 x$

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

Differentiate.

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Step 1.1.1.1.1

By the Sum Rule, the derivative of $x^{3} - 6 x^{2} + 9 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [9 x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [9 x]$

Step 1.1.1.1.2

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

$3 x^{2} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [9 x]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [- 6 x^{2}]$ .

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Step 1.1.1.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - 6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9 x]$

Step 1.1.1.2.2

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

$3 x^{2} - 6 (2 x) + \frac{d}{d x} [9 x]$

Step 1.1.1.2.3

Multiply $2$ by $- 6$ .

$3 x^{2} - 12 x + \frac{d}{d x} [9 x]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 1.1.1.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$3 x^{2} - 12 x + 9 \frac{d}{d x} [x]$

Step 1.1.1.3.2

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

$3 x^{2} - 12 x + 9 \cdot 1$

Step 1.1.1.3.3

Multiply $9$ by $1$ .

$f' (x) = 3 x^{2} - 12 x + 9$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $3 x^{2} - 12 x + 9$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [9]$

Step 1.1.2.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 1.1.2.2.1

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

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [9]$

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 $n = 2$ .

$3 (2 x) + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [9]$

Step 1.1.2.2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [9]$

Step 1.1.2.3

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

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Step 1.1.2.3.1

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

$6 x - 12 \frac{d}{d x} [x] + \frac{d}{d x} [9]$

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 $n = 1$ .

$6 x - 12 \cdot 1 + \frac{d}{d x} [9]$

Step 1.1.2.3.3

Multiply $- 12$ by $1$ .

$6 x - 12 + \frac{d}{d x} [9]$

Step 1.1.2.4

Differentiate using the Constant Rule.

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Step 1.1.2.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$6 x - 12 + 0$

Step 1.1.2.4.2

Add $6 x - 12$ and $0$ .

$f'' (x) = 6 x - 12$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $6 x - 12$ .

$6 x - 12$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $6 x - 12 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$6 x - 12 = 0$

Step 1.2.2

Add $12$ to both sides of the equation.

$6 x = 12$

Step 1.2.3

Divide each term in $6 x = 12$ by $6$ and simplify.

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Step 1.2.3.1

Divide each term in $6 x = 12$ by $6$ .

$\frac{6 x}{6} = \frac{12}{6}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $6$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{12}{6}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{6}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $12$ by $6$ .

$x = 2$

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.

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

Step 4

Substitute any number from the interval $(- \infty, 2)$ 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.

$f'' (0) = 6 (0) - 12$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $6$ by $0$ .

$f'' (0) = 0 - 12$

Step 4.2.2

Subtract $12$ from $0$ .

$f'' (0) = - 12$

Step 4.2.3

The final answer is $- 12$ .

$- 12$

Step 4.3

The graph is concave down on the interval $(- \infty, 2)$ because $f'' (0)$ is negative.

Concave down on $(- \infty, 2)$ since $f'' (x)$ is negative

Step 5

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

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Step 5.1

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

$f'' (4) = 6 (4) - 12$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $6$ by $4$ .

$f'' (4) = 24 - 12$

Step 5.2.2

Subtract $12$ from $24$ .

$f'' (4) = 12$

Step 5.2.3

The final answer is $12$ .

$12$

Step 5.3

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 6

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

Concave down on $(- \infty, 2)$ since $f'' (x)$ is negative

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

Step 7