Calculus Examples

$f (x) = - 6 x$

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

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

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

Step 1.1.1.2

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

$- 6 \cdot 1$

Step 1.1.1.3

Multiply $- 6$ by $1$ .

$f' (x) = - 6$

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