Calculus Examples

$f (x) = x^{2} + 10 x - 9$

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

Differentiate.

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Step 1.1.1.1.1

By the Sum Rule, the derivative of $x^{2} + 10 x - 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [- 9]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [- 9]$

Step 1.1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [10 x] + \frac{d}{d x} [- 9]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [10 x]$ .

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Step 1.1.1.2.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$2 x + 10 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]$

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 $n = 1$ .

$2 x + 10 \cdot 1 + \frac{d}{d x} [- 9]$

Step 1.1.1.2.3

Multiply $10$ by $1$ .

$2 x + 10 + \frac{d}{d x} [- 9]$

Step 1.1.1.3

Differentiate using the Constant Rule.

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Step 1.1.1.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$2 x + 10 + 0$

Step 1.1.1.3.2

Add $2 x + 10$ and $0$ .

$f' (x) = 2 x + 10$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $2 x + 10$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [10]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [10]$

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [10]$

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 $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [10]$

Step 1.1.2.2.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [10]$

Step 1.1.2.3

Differentiate using the Constant Rule.

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Step 1.1.2.3.1

Since $10$ is constant with respect to $x$ , the derivative of $10$ 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 up because the second derivative is positive.

The graph is concave up

Step 4