Calculus Examples

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

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

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

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

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 $n = 2$ .

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

Step 1.1.1.2.3

Multiply $2$ by $- 1$ .

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

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

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 $n = 1$ .

$- 2 x + 2 \cdot 1 + \frac{d}{d x} [6]$

Step 1.1.1.3.3

Multiply $2$ by $1$ .

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

Step 1.1.1.4

Differentiate using the Constant Rule.

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Step 1.1.1.4.1

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

$- 2 x + 2 + 0$

Step 1.1.1.4.2

Add $- 2 x + 2$ and $0$ .

$f' (x) = - 2 x + 2$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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Step 1.1.2.2.1

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

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

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 $n = 1$ .

$- 2 \cdot 1 + \frac{d}{d x} [2]$

Step 1.1.2.2.3

Multiply $- 2$ by $1$ .

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

Step 1.1.2.3

Differentiate using the Constant Rule.

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Step 1.1.2.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$- 2 + 0$

Step 1.1.2.3.2

Add $- 2$ and $0$ .

$f'' (x) = - 2$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $- 2$ .

$- 2$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- 2 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- 2 = 0$

Step 1.2.2

Since $- 2 \neq 0$ , there are no solutions.

No solution

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

The graph is concave down because the second derivative is negative.

The graph is concave down

Step 4

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