Calculus Examples

$f (x) = 5 x^{2} + 8 x - 6$

Step 1

Find the first derivative of the function.

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By the Sum Rule, the derivative of $5 x^{2} + 8 x - 6$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [- 6]$ .

$\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [- 6]$

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [- 6]$

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

$5 (2 x) + \frac{d}{d x} [8 x] + \frac{d}{d x} [- 6]$

Multiply $2$ by $5$ .

$10 x + \frac{d}{d x} [8 x] + \frac{d}{d x} [- 6]$

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

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

$10 x + 8 \frac{d}{d x} [x] + \frac{d}{d x} [- 6]$

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

$10 x + 8 \cdot 1 + \frac{d}{d x} [- 6]$

Multiply $8$ by $1$ .

$10 x + 8 + \frac{d}{d x} [- 6]$

Differentiate using the Constant Rule.

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

$10 x + 8 + 0$

Add $10 x + 8$ and $0$ .

$10 x + 8$

Step 2

Find the second derivative of the function.

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By the Sum Rule, the derivative of $10 x + 8$ with respect to $x$ is $\frac{d}{d x} [10 x] + \frac{d}{d x} [8]$ .

$f'' (x) = \frac{d}{d x} (10 x) + \frac{d}{d x} (8)$

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

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

$f'' (x) = 10 \frac{d}{d x} (x) + \frac{d}{d x} (8)$

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

$f'' (x) = 10 \cdot 1 + \frac{d}{d x} (8)$

Multiply $10$ by $1$ .

$f'' (x) = 10 + \frac{d}{d x} (8)$

Differentiate using the Constant Rule.

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Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$f'' (x) = 10 + 0$

Add $10$ and $0$ .

$f'' (x) = 10$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$10 x + 8 = 0$

Step 4

Find the first derivative.

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Find the first derivative.

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By the Sum Rule, the derivative of $5 x^{2} + 8 x - 6$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [- 6]$ .

$f' (x) = \frac{d}{d x} (5 x^{2}) + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 6)$

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 5 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 6)$

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

$f' (x) = 5 (2 x) + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 6)$

Multiply $2$ by $5$ .

$f' (x) = 10 x + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 6)$

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

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

$f' (x) = 10 x + 8 \frac{d}{d x} (x) + \frac{d}{d x} (- 6)$

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

$f' (x) = 10 x + 8 \cdot 1 + \frac{d}{d x} (- 6)$

Multiply $8$ by $1$ .

$f' (x) = 10 x + 8 + \frac{d}{d x} (- 6)$

Differentiate using the Constant Rule.

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

$f' (x) = 10 x + 8 + 0$

Add $10 x + 8$ and $0$ .

$f' (x) = 10 x + 8$

The first derivative of $f (x)$ with respect to $x$ is $10 x + 8$ .

$10 x + 8$

Step 5

Set the first derivative equal to $0$ then solve the equation $10 x + 8 = 0$ .

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Set the first derivative equal to $0$ .

$10 x + 8 = 0$

Subtract $8$ from both sides of the equation.

$10 x = - 8$

Divide each term in $10 x = - 8$ by $10$ and simplify.

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Divide each term in $10 x = - 8$ by $10$ .

$\frac{10 x}{10} = \frac{- 8}{10}$

Simplify the left side.

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Cancel the common factor of $10$ .

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Cancel the common factor.

$\frac{10 x}{10} = \frac{- 8}{10}$

Divide $x$ by $1$ .

$x = \frac{- 8}{10}$

Simplify the right side.

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Cancel the common factor of $- 8$ and $10$ .

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Factor $2$ out of $- 8$ .

$x = \frac{2 (- 4)}{10}$

Cancel the common factors.

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Factor $2$ out of $10$ .

$x = \frac{2 \cdot - 4}{2 \cdot 5}$

Cancel the common factor.

$x = \frac{2 \cdot - 4}{2 \cdot 5}$

Rewrite the expression.

$x = \frac{- 4}{5}$

Move the negative in front of the fraction.

$x = - \frac{4}{5}$

Step 6

Find the values where the derivative is undefined.

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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.

Step 7

Critical points to evaluate.

$x = - \frac{4}{5}$

Step 8

Evaluate the second derivative at $x = - \frac{4}{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$10$

Step 9

$x = - \frac{4}{5}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{4}{5}$ is a local minimum

Step 10

Find the y-value when $x = - \frac{4}{5}$ .

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Replace the variable $x$ with $- \frac{4}{5}$ in the expression.

$f (- \frac{4}{5}) = 5 {(- \frac{4}{5})}^{2} + 8 (- \frac{4}{5}) - 6$

Simplify the result.

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Simplify each term.

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Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{4}{5}$ .

$f (- \frac{4}{5}) = 5 ({(- 1)}^{2} {(\frac{4}{5})}^{2}) + 8 (- \frac{4}{5}) - 6$

Apply the product rule to $\frac{4}{5}$ .

$f (- \frac{4}{5}) = 5 ({(- 1)}^{2} (\frac{4^{2}}{5^{2}})) + 8 (- \frac{4}{5}) - 6$

Raise $- 1$ to the power of $2$ .

$f (- \frac{4}{5}) = 5 (1 (\frac{4^{2}}{5^{2}})) + 8 (- \frac{4}{5}) - 6$

Multiply $\frac{4^{2}}{5^{2}}$ by $1$ .

$f (- \frac{4}{5}) = 5 (\frac{4^{2}}{5^{2}}) + 8 (- \frac{4}{5}) - 6$

Raise $4$ to the power of $2$ .

$f (- \frac{4}{5}) = 5 (\frac{16}{5^{2}}) + 8 (- \frac{4}{5}) - 6$

Raise $5$ to the power of $2$ .

$f (- \frac{4}{5}) = 5 (\frac{16}{25}) + 8 (- \frac{4}{5}) - 6$

Cancel the common factor of $5$ .

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Factor $5$ out of $25$ .

$f (- \frac{4}{5}) = 5 (\frac{16}{5 (5)}) + 8 (- \frac{4}{5}) - 6$

Cancel the common factor.

$f (- \frac{4}{5}) = 5 (\frac{16}{5 \cdot 5}) + 8 (- \frac{4}{5}) - 6$

Rewrite the expression.

$f (- \frac{4}{5}) = \frac{16}{5} + 8 (- \frac{4}{5}) - 6$

Multiply $8 (- \frac{4}{5})$ .

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Multiply $- 1$ by $8$ .

$f (- \frac{4}{5}) = \frac{16}{5} - 8 (\frac{4}{5}) - 6$

Combine $- 8$ and $\frac{4}{5}$ .

$f (- \frac{4}{5}) = \frac{16}{5} + \frac{- 8 \cdot 4}{5} - 6$

Multiply $- 8$ by $4$ .

$f (- \frac{4}{5}) = \frac{16}{5} + \frac{- 32}{5} - 6$

Move the negative in front of the fraction.

$f (- \frac{4}{5}) = \frac{16}{5} - \frac{32}{5} - 6$

Combine fractions.

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Combine the numerators over the common denominator.

$f (- \frac{4}{5}) = - 6 + \frac{16 - 32}{5}$

Simplify the expression.

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Subtract $32$ from $16$ .

$f (- \frac{4}{5}) = - 6 + \frac{- 16}{5}$

Move the negative in front of the fraction.

$f (- \frac{4}{5}) = - 6 - \frac{16}{5}$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (- \frac{4}{5}) = - 6 \cdot \frac{5}{5} - \frac{16}{5}$

Combine $- 6$ and $\frac{5}{5}$ .

$f (- \frac{4}{5}) = \frac{- 6 \cdot 5}{5} - \frac{16}{5}$

Combine the numerators over the common denominator.

$f (- \frac{4}{5}) = \frac{- 6 \cdot 5 - 16}{5}$

Simplify the numerator.

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Multiply $- 6$ by $5$ .

$f (- \frac{4}{5}) = \frac{- 30 - 16}{5}$

Subtract $16$ from $- 30$ .

$f (- \frac{4}{5}) = \frac{- 46}{5}$

Move the negative in front of the fraction.

$f (- \frac{4}{5}) = - \frac{46}{5}$

The final answer is $- \frac{46}{5}$ .

$y = - \frac{46}{5}$

Step 11

These are the local extrema for $f (x) = 5 x^{2} + 8 x - 6$ .

$(- \frac{4}{5}, - \frac{46}{5})$ is a local minima

Step 12