Calculus Examples

$y = 4 x^{2} - x - 6$

Step 1

Write $y = 4 x^{2} - x - 6$ as a function.

$f (x) = 4 x^{2} - x - 6$

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

By the Sum Rule, the derivative of $4 x^{2} - x - 6$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$ .

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$

Step 2.1.1.2

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 2.1.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$

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 $n = 2$ .

$4 (2 x) + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$

Step 2.1.1.2.3

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$

Step 2.1.1.3

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

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Step 2.1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$8 x - \frac{d}{d x} [x] + \frac{d}{d x} [- 6]$

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 $n = 1$ .

$8 x - 1 \cdot 1 + \frac{d}{d x} [- 6]$

Step 2.1.1.3.3

Multiply $- 1$ by $1$ .

$8 x - 1 + \frac{d}{d x} [- 6]$

Step 2.1.1.4

Differentiate using the Constant Rule.

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Step 2.1.1.4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$8 x - 1 + 0$

Step 2.1.1.4.2

Add $8 x - 1$ and $0$ .

$f' (x) = 8 x - 1$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

By the Sum Rule, the derivative of $8 x - 1$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [8 x] + \frac{d}{d x} [- 1]$

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

$8 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

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 $n = 1$ .

$8 \cdot 1 + \frac{d}{d x} [- 1]$

Step 2.1.2.2.3

Multiply $8$ by $1$ .

$8 + \frac{d}{d x} [- 1]$

Step 2.1.2.3

Differentiate using the Constant Rule.

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Step 2.1.2.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$8 + 0$

Step 2.1.2.3.2

Add $8$ and $0$ .

$f'' (x) = 8$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $8$ .

$8$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $8 = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$8 = 0$

Step 2.2.2

Since $8 \neq 0$ , there are no solutions.

No solution

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

The graph is concave up because the second derivative is positive.

The graph is concave up

Step 5