Calculus Examples

f (x) = - 6 x

$f (x) = - 6 x$

Step 1

Find the

x

$x$ values where the second derivative is equal to

0

$0$ .

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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

- 6

$- 6$ is constant with respect to

x

$x$ , the derivative of

- 6 x

$- 6 x$ with respect to

x

$x$ is

- 6 \frac{d}{d x} [x]

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

- 6 \frac{d}{d x} [x]

$- 6 \frac{d}{d x} [x]$

Step 1.1.1.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

- 6 \cdot 1

$- 6 \cdot 1$

Step 1.1.1.3

Multiply

- 6

$- 6$ by

1

$1$ .

$f' (x) = - 6$

Step 1.1.2

Since

$- 6$ is constant with respect to

$x$ , the derivative of

$- 6$ with respect to

$x$ is

$0$ .

$f'' (x) = 0$

Step 1.1.3

The second derivative of

$f (x)$ with respect to

$x$ is

$0$ .

$0$

Step 1.2

Set the second derivative equal to

$0$ then solve the equation

$0 = 0$ .

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Step 1.2.1

Set the second derivative equal to

$0$ .

$0 = 0$

Step 1.2.2

Since

$0 = 0$ , the equation will always be true.

Always true

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.

$(- \infty, \infty)$

Step 4

Substitute any number from the interval

$(- \infty, \infty)$ 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.

$f'' (0) = 0$

Step 4.2

The final answer is

$0$ .

$0$

Step 4.3

The graph is concave up on the interval

$(- \infty, \infty)$ because

$f'' (0)$ is positive.

The graph is concave up

Step 5