Calculus Examples

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

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

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

By the Sum Rule, the derivative of

- x^{2} + 2 x + 6

$- x^{2} + 2 x + 6$ with respect to

x

$x$ is

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

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

\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}]

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

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Step 1.1.1.2.1

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- x^{2}

$- x^{2}$ with respect to

x

$x$ is

- \frac{d}{d x} [x^{2}]

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

- \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 1.1.1.2.2

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 = 2

$n = 2$ .

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

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

Step 1.1.1.2.3

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

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

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

Step 1.1.1.3

Evaluate

\frac{d}{d x} [2 x]

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

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Step 1.1.1.3.1

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2 x

$2 x$ with respect to

x

$x$ is

2 \frac{d}{d x} [x]

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

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

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

Step 1.1.1.3.2

Differentiate using the Power Rule which states that

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

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

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

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

Step 1.1.1.3.3

Multiply

2

$2$ by

1

$1$ .

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

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

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

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

Step 1.1.1.4

Differentiate using the Constant Rule.

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Step 1.1.1.4.1

Since

6

$6$ is constant with respect to

x

$x$ , the derivative of

6

$6$ with respect to

x

$x$ is

0

$0$ .

- 2 x + 2 + 0

$- 2 x + 2 + 0$

Step 1.1.1.4.2

Add

- 2 x + 2

$- 2 x + 2$ and

0

$0$ .

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

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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