Calculus Examples

$7 t + \frac{3}{t}$

Step 1

Write $7 t + \frac{3}{t}$ as a function.

$f (t) = 7 t + \frac{3}{t}$

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $7 t + \frac{3}{t}$ with respect to $t$ is $\frac{d}{d t} [7 t] + \frac{d}{d t} [\frac{3}{t}]$ .

$\frac{d}{d t} [7 t] + \frac{d}{d t} [\frac{3}{t}]$

Step 2.2

Evaluate $\frac{d}{d t} [7 t]$ .

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Step 2.2.1

Since $7$ is constant with respect to $t$ , the derivative of $7 t$ with respect to $t$ is $7 \frac{d}{d t} [t]$ .

$7 \frac{d}{d t} [t] + \frac{d}{d t} [\frac{3}{t}]$

Step 2.2.2

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

$7 \cdot 1 + \frac{d}{d t} [\frac{3}{t}]$

Step 2.2.3

Multiply $7$ by $1$ .

$7 + \frac{d}{d t} [\frac{3}{t}]$

Step 2.3

Evaluate $\frac{d}{d t} [\frac{3}{t}]$ .

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Step 2.3.1

Since $3$ is constant with respect to $t$ , the derivative of $\frac{3}{t}$ with respect to $t$ is $3 \frac{d}{d t} [\frac{1}{t}]$ .

$7 + 3 \frac{d}{d t} [\frac{1}{t}]$

Step 2.3.2

Rewrite $\frac{1}{t}$ as $t^{- 1}$ .

$7 + 3 \frac{d}{d t} [t^{- 1}]$

Step 2.3.3

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

$7 + 3 (- t^{- 2})$

Step 2.3.4

Multiply $- 1$ by $3$ .

$7 - 3 t^{- 2}$

Step 2.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$7 - 3 \frac{1}{t^{2}}$

Step 2.5

Simplify.

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Step 2.5.1

Combine terms.

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Step 2.5.1.1

Combine $- 3$ and $\frac{1}{t^{2}}$ .

$7 + \frac{- 3}{t^{2}}$

Step 2.5.1.2

Move the negative in front of the fraction.

$7 - \frac{3}{t^{2}}$

Step 2.5.2

Reorder terms.

$- \frac{3}{t^{2}} + 7$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $- \frac{3}{t^{2}} + 7$ with respect to $t$ is $\frac{d}{d t} [- \frac{3}{t^{2}}] + \frac{d}{d t} [7]$ .

$f'' (t) = \frac{d}{d t} (- \frac{3}{t^{2}}) + \frac{d}{d t} (7)$

Step 3.2

Evaluate $\frac{d}{d t} [- \frac{3}{t^{2}}]$ .

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Step 3.2.1

Since $- 3$ is constant with respect to $t$ , the derivative of $- \frac{3}{t^{2}}$ with respect to $t$ is $- 3 \frac{d}{d t} [\frac{1}{t^{2}}]$ .

$f'' (t) = - 3 \frac{d}{d t} \frac{1}{t^{2}} + \frac{d}{d t} (7)$

Step 3.2.2

Rewrite $\frac{1}{t^{2}}$ as ${(t^{2})}^{- 1}$ .

$f'' (t) = - 3 \frac{d}{d t} {(t^{2})}^{- 1} + \frac{d}{d t} (7)$

Step 3.2.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = t^{2}$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u$ as $t^{2}$ .

$f'' (t) = - 3 (\frac{d}{d u} (u^{- 1}) \frac{d}{d t} (t^{2})) + \frac{d}{d t} (7)$

Step 3.2.3.2

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

$f'' (t) = - 3 (- u^{- 2} \frac{d}{d t} t^{2}) + \frac{d}{d t} (7)$

Step 3.2.3.3

Replace all occurrences of $u$ with $t^{2}$ .

$f'' (t) = - 3 (- {(t^{2})}^{- 2} \frac{d}{d t} t^{2}) + \frac{d}{d t} (7)$

Step 3.2.4

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

$f'' (t) = - 3 (- {(t^{2})}^{- 2} (2 t)) + \frac{d}{d t} (7)$

Step 3.2.5

Multiply the exponents in ${(t^{2})}^{- 2}$ .

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Step 3.2.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (t) = - 3 (- t^{2 \cdot - 2} (2 t)) + \frac{d}{d t} (7)$

Step 3.2.5.2

Multiply $2$ by $- 2$ .

$f'' (t) = - 3 (- t^{- 4} (2 t)) + \frac{d}{d t} (7)$

Step 3.2.6

Multiply $2$ by $- 1$ .

$f'' (t) = - 3 (- 2 t^{- 4} t) + \frac{d}{d t} (7)$

Step 3.2.7

Raise $t$ to the power of $1$ .

$f'' (t) = - 3 (- 2 (t \cdot t^{- 4})) + \frac{d}{d t} (7)$

Step 3.2.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (t) = - 3 (- 2 t^{1 - 4}) + \frac{d}{d t} (7)$

Step 3.2.9

Subtract $4$ from $1$ .

$f'' (t) = - 3 (- 2 t^{- 3}) + \frac{d}{d t} (7)$

Step 3.2.10

Multiply $- 2$ by $- 3$ .

$f'' (t) = 6 t^{- 3} + \frac{d}{d t} (7)$

Step 3.3

Since $7$ is constant with respect to $t$ , the derivative of $7$ with respect to $t$ is $0$ .

$f'' (t) = 6 t^{- 3} + 0$

Step 3.4

Simplify.

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Step 3.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (t) = 6 (\frac{1}{t^{3}}) + 0$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $6$ and $\frac{1}{t^{3}}$ .

$f'' (t) = \frac{6}{t^{3}} + 0$

Step 3.4.2.2

Add $\frac{6}{t^{3}}$ and $0$ .

$f'' (t) = \frac{6}{t^{3}}$

Step 4

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

$- \frac{3}{t^{2}} + 7 = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

By the Sum Rule, the derivative of $7 t + \frac{3}{t}$ with respect to $t$ is $\frac{d}{d t} [7 t] + \frac{d}{d t} [\frac{3}{t}]$ .

$\frac{d}{d t} [7 t] + \frac{d}{d t} [\frac{3}{t}]$

Step 5.1.2

Evaluate $\frac{d}{d t} [7 t]$ .

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Step 5.1.2.1

Since $7$ is constant with respect to $t$ , the derivative of $7 t$ with respect to $t$ is $7 \frac{d}{d t} [t]$ .

$7 \frac{d}{d t} [t] + \frac{d}{d t} [\frac{3}{t}]$

Step 5.1.2.2

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

$7 \cdot 1 + \frac{d}{d t} [\frac{3}{t}]$

Step 5.1.2.3

Multiply $7$ by $1$ .

$7 + \frac{d}{d t} [\frac{3}{t}]$

Step 5.1.3

Evaluate $\frac{d}{d t} [\frac{3}{t}]$ .

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Step 5.1.3.1

Since $3$ is constant with respect to $t$ , the derivative of $\frac{3}{t}$ with respect to $t$ is $3 \frac{d}{d t} [\frac{1}{t}]$ .

$7 + 3 \frac{d}{d t} [\frac{1}{t}]$

Step 5.1.3.2

Rewrite $\frac{1}{t}$ as $t^{- 1}$ .

$7 + 3 \frac{d}{d t} [t^{- 1}]$

Step 5.1.3.3

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

$7 + 3 (- t^{- 2})$

Step 5.1.3.4

Multiply $- 1$ by $3$ .

$7 - 3 t^{- 2}$

Step 5.1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$7 - 3 \frac{1}{t^{2}}$

Step 5.1.5

Simplify.

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Step 5.1.5.1

Combine terms.

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Step 5.1.5.1.1

Combine $- 3$ and $\frac{1}{t^{2}}$ .

$7 + \frac{- 3}{t^{2}}$

Step 5.1.5.1.2

Move the negative in front of the fraction.

$7 - \frac{3}{t^{2}}$

Step 5.1.5.2

Reorder terms.

$f' (t) = - \frac{3}{t^{2}} + 7$

Step 5.2

The first derivative of $f (t)$ with respect to $t$ is $- \frac{3}{t^{2}} + 7$ .

$- \frac{3}{t^{2}} + 7$

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{3}{t^{2}} + 7 = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- \frac{3}{t^{2}} + 7 = 0$

Step 6.2

Subtract $7$ from both sides of the equation.

$- \frac{3}{t^{2}} = - 7$

Step 6.3

Find the LCD of the terms in the equation.

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Step 6.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$t^{2}, 1$

Step 6.3.2

The LCM of one and any expression is the expression.

$t^{2}$

Step 6.4

Multiply each term in $- \frac{3}{t^{2}} = - 7$ by $t^{2}$ to eliminate the fractions.

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Step 6.4.1

Multiply each term in $- \frac{3}{t^{2}} = - 7$ by $t^{2}$ .

$- \frac{3}{t^{2}} t^{2} = - 7 t^{2}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $t^{2}$ .

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Step 6.4.2.1.1

Move the leading negative in $- \frac{3}{t^{2}}$ into the numerator.

$\frac{- 3}{t^{2}} t^{2} = - 7 t^{2}$

Step 6.4.2.1.2

Cancel the common factor.

$\frac{- 3}{t^{2}} t^{2} = - 7 t^{2}$

Step 6.4.2.1.3

Rewrite the expression.

$- 3 = - 7 t^{2}$

Step 6.5

Solve the equation.

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Step 6.5.1

Rewrite the equation as $- 7 t^{2} = - 3$ .

$- 7 t^{2} = - 3$

Step 6.5.2

Divide each term in $- 7 t^{2} = - 3$ by $- 7$ and simplify.

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Step 6.5.2.1

Divide each term in $- 7 t^{2} = - 3$ by $- 7$ .

$\frac{- 7 t^{2}}{- 7} = \frac{- 3}{- 7}$

Step 6.5.2.2

Simplify the left side.

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Step 6.5.2.2.1

Cancel the common factor of $- 7$ .

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Step 6.5.2.2.1.1

Cancel the common factor.

$\frac{- 7 t^{2}}{- 7} = \frac{- 3}{- 7}$

Step 6.5.2.2.1.2

Divide $t^{2}$ by $1$ .

$t^{2} = \frac{- 3}{- 7}$

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

Dividing two negative values results in a positive value.

$t^{2} = \frac{3}{7}$

Step 6.5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{\frac{3}{7}}$

Step 6.5.4

Simplify $\pm \sqrt{\frac{3}{7}}$ .

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Step 6.5.4.1

Rewrite $\sqrt{\frac{3}{7}}$ as $\frac{\sqrt{3}}{\sqrt{7}}$ .

$t = \pm \frac{\sqrt{3}}{\sqrt{7}}$

Step 6.5.4.2

Multiply $\frac{\sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$t = \pm \frac{\sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 6.5.4.3

Combine and simplify the denominator.

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Step 6.5.4.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$t = \pm \frac{\sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 6.5.4.3.2

Raise $\sqrt{7}$ to the power of $1$ .

$t = \pm \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 6.5.4.3.3

Raise $\sqrt{7}$ to the power of $1$ .

$t = \pm \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 6.5.4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$t = \pm \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 6.5.4.3.5

Add $1$ and $1$ .

$t = \pm \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 6.5.4.3.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 6.5.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$t = \pm \frac{\sqrt{3} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 6.5.4.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$t = \pm \frac{\sqrt{3} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 6.5.4.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$t = \pm \frac{\sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 6.5.4.3.6.4

Cancel the common factor of $2$ .

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Step 6.5.4.3.6.4.1

Cancel the common factor.

$t = \pm \frac{\sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 6.5.4.3.6.4.2

Rewrite the expression.

$t = \pm \frac{\sqrt{3} \sqrt{7}}{7^{1}}$

Step 6.5.4.3.6.5

Evaluate the exponent.

$t = \pm \frac{\sqrt{3} \sqrt{7}}{7}$

Step 6.5.4.4

Simplify the numerator.

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Step 6.5.4.4.1

Combine using the product rule for radicals.

$t = \pm \frac{\sqrt{3 \cdot 7}}{7}$

Step 6.5.4.4.2

Multiply $3$ by $7$ .

$t = \pm \frac{\sqrt{21}}{7}$

Step 6.5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$t = \frac{\sqrt{21}}{7}$

Step 6.5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$t = - \frac{\sqrt{21}}{7}$

Step 6.5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$t = \frac{\sqrt{21}}{7}, - \frac{\sqrt{21}}{7}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{3}{t^{2}}$ equal to $0$ to find where the expression is undefined.

$t^{2} = 0$

Step 7.2

Solve for $t$ .

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Step 7.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{0}$

Step 7.2.2

Simplify $\pm \sqrt{0}$ .

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Step 7.2.2.1

Rewrite $0$ as $0^{2}$ .

$t = \pm \sqrt{0^{2}}$

Step 7.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$t = \pm 0$

Step 7.2.2.3

Plus or minus $0$ is $0$ .

$t = 0$

Step 8

Critical points to evaluate.

$t = \frac{\sqrt{21}}{7}, - \frac{\sqrt{21}}{7}$

Step 9

Evaluate the second derivative at $t = \frac{\sqrt{21}}{7}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{6}{{(\frac{\sqrt{21}}{7})}^{3}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the denominator.

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Step 10.1.1

Apply the product rule to $\frac{\sqrt{21}}{7}$ .

$\frac{6}{\frac{{\sqrt{21}}^{3}}{7^{3}}}$

Step 10.1.2

Simplify the numerator.

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Step 10.1.2.1

Rewrite ${\sqrt{21}}^{3}$ as $\sqrt{21^{3}}$ .

$\frac{6}{\frac{\sqrt{21^{3}}}{7^{3}}}$

Step 10.1.2.2

Raise $21$ to the power of $3$ .

$\frac{6}{\frac{\sqrt{9261}}{7^{3}}}$

Step 10.1.2.3

Rewrite $9261$ as $21^{2} \cdot 21$ .

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Step 10.1.2.3.1

Factor $441$ out of $9261$ .

$\frac{6}{\frac{\sqrt{441 (21)}}{7^{3}}}$

Step 10.1.2.3.2

Rewrite $441$ as $21^{2}$ .

$\frac{6}{\frac{\sqrt{21^{2} \cdot 21}}{7^{3}}}$

Step 10.1.2.4

Pull terms out from under the radical.

$\frac{6}{\frac{21 \sqrt{21}}{7^{3}}}$

Step 10.1.3

Raise $7$ to the power of $3$ .

$\frac{6}{\frac{21 \sqrt{21}}{343}}$

Step 10.1.4

Cancel the common factor of $21$ and $343$ .

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Step 10.1.4.1

Factor $7$ out of $21 \sqrt{21}$ .

$\frac{6}{\frac{7 (3 \sqrt{21})}{343}}$

Step 10.1.4.2

Cancel the common factors.

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Step 10.1.4.2.1

Factor $7$ out of $343$ .

$\frac{6}{\frac{7 (3 \sqrt{21})}{7 (49)}}$

Step 10.1.4.2.2

Cancel the common factor.

$\frac{6}{\frac{7 (3 \sqrt{21})}{7 \cdot 49}}$

Step 10.1.4.2.3

Rewrite the expression.

$\frac{6}{\frac{3 \sqrt{21}}{49}}$

Step 10.2

Multiply the numerator by the reciprocal of the denominator.

$6 \frac{49}{3 \sqrt{21}}$

Step 10.3

Cancel the common factor of $3$ .

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Step 10.3.1

Factor $3$ out of $6$ .

$3 (2) \frac{49}{3 \sqrt{21}}$

Step 10.3.2

Factor $3$ out of $3 \sqrt{21}$ .

$3 (2) \frac{49}{3 (\sqrt{21})}$

Step 10.3.3

Cancel the common factor.

$3 \cdot 2 \frac{49}{3 \sqrt{21}}$

Step 10.3.4

Rewrite the expression.

$2 \frac{49}{\sqrt{21}}$

Step 10.4

Combine $2$ and $\frac{49}{\sqrt{21}}$ .

$\frac{2 \cdot 49}{\sqrt{21}}$

Step 10.5

Multiply $2$ by $49$ .

$\frac{98}{\sqrt{21}}$

Step 10.6

Multiply $\frac{98}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\frac{98}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}}$

Step 10.7

Combine and simplify the denominator.

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Step 10.7.1

Multiply $\frac{98}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\frac{98 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Step 10.7.2

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{98 \sqrt{21}}{{\sqrt{21}}^{1} \sqrt{21}}$

Step 10.7.3

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{98 \sqrt{21}}{{\sqrt{21}}^{1} {\sqrt{21}}^{1}}$

Step 10.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{98 \sqrt{21}}{{\sqrt{21}}^{1 + 1}}$

Step 10.7.5

Add $1$ and $1$ .

$\frac{98 \sqrt{21}}{{\sqrt{21}}^{2}}$

Step 10.7.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 10.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\frac{98 \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}}$

Step 10.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{98 \sqrt{21}}{21^{\frac{1}{2} \cdot 2}}$

Step 10.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{98 \sqrt{21}}{21^{\frac{2}{2}}}$

Step 10.7.6.4

Cancel the common factor of $2$ .

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Step 10.7.6.4.1

Cancel the common factor.

$\frac{98 \sqrt{21}}{21^{\frac{2}{2}}}$

Step 10.7.6.4.2

Rewrite the expression.

$\frac{98 \sqrt{21}}{21^{1}}$

Step 10.7.6.5

Evaluate the exponent.

$\frac{98 \sqrt{21}}{21}$

Step 10.8

Cancel the common factor of $98$ and $21$ .

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Step 10.8.1

Factor $7$ out of $98 \sqrt{21}$ .

$\frac{7 (14 \sqrt{21})}{21}$

Step 10.8.2

Cancel the common factors.

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Step 10.8.2.1

Factor $7$ out of $21$ .

$\frac{7 (14 \sqrt{21})}{7 (3)}$

Step 10.8.2.2

Cancel the common factor.

$\frac{7 (14 \sqrt{21})}{7 \cdot 3}$

Step 10.8.2.3

Rewrite the expression.

$\frac{14 \sqrt{21}}{3}$

Step 11

$t = \frac{\sqrt{21}}{7}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$t = \frac{\sqrt{21}}{7}$ is a local minimum

Step 12

Find the y-value when $t = \frac{\sqrt{21}}{7}$ .

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Step 12.1

Replace the variable $t$ with $\frac{\sqrt{21}}{7}$ in the expression.

$f (\frac{\sqrt{21}}{7}) = 7 (\frac{\sqrt{21}}{7}) + \frac{3}{\frac{\sqrt{21}}{7}}$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Cancel the common factor of $7$ .

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Step 12.2.1.1.1

Cancel the common factor.

$f (\frac{\sqrt{21}}{7}) = 7 (\frac{\sqrt{21}}{7}) + \frac{3}{\frac{\sqrt{21}}{7}}$

Step 12.2.1.1.2

Rewrite the expression.

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + \frac{3}{\frac{\sqrt{21}}{7}}$

Step 12.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7}{\sqrt{21}})$

Step 12.2.1.3

Multiply $\frac{7}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}})$

Step 12.2.1.4

Combine and simplify the denominator.

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Step 12.2.1.4.1

Multiply $\frac{7}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7 \sqrt{21}}{\sqrt{21} \sqrt{21}})$

Step 12.2.1.4.2

Raise $\sqrt{21}$ to the power of $1$ .

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7 \sqrt{21}}{\sqrt{21} \sqrt{21}})$

Step 12.2.1.4.3

Raise $\sqrt{21}$ to the power of $1$ .

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7 \sqrt{21}}{\sqrt{21} \sqrt{21}})$

Step 12.2.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7 \sqrt{21}}{{\sqrt{21}}^{1 + 1}})$

Step 12.2.1.4.5

Add $1$ and $1$ .

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7 \sqrt{21}}{{\sqrt{21}}^{2}})$

Step 12.2.1.4.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 12.2.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7 \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}})$

Step 12.2.1.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7 \sqrt{21}}{21^{\frac{1}{2} \cdot 2}})$

Step 12.2.1.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7 \sqrt{21}}{21^{\frac{2}{2}}})$

Step 12.2.1.4.6.4

Cancel the common factor of $2$ .

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Step 12.2.1.4.6.4.1

Cancel the common factor.

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7 \sqrt{21}}{21^{\frac{2}{2}}})$

Step 12.2.1.4.6.4.2

Rewrite the expression.

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7 \sqrt{21}}{21})$

Step 12.2.1.4.6.5

Evaluate the exponent.

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7 \sqrt{21}}{21})$

Step 12.2.1.5

Cancel the common factor of $3$ .

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Step 12.2.1.5.1

Factor $3$ out of $21$ .

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7 \sqrt{21}}{3 (7)})$

Step 12.2.1.5.2

Cancel the common factor.

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + 3 (\frac{7 \sqrt{21}}{3 \cdot 7})$

Step 12.2.1.5.3

Rewrite the expression.

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + \frac{7 \sqrt{21}}{7}$

Step 12.2.1.6

Cancel the common factor of $7$ .

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Step 12.2.1.6.1

Cancel the common factor.

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + \frac{7 \sqrt{21}}{7}$

Step 12.2.1.6.2

Divide $\sqrt{21}$ by $1$ .

$f (\frac{\sqrt{21}}{7}) = \sqrt{21} + \sqrt{21}$

Step 12.2.2

Add $\sqrt{21}$ and $\sqrt{21}$ .

$f (\frac{\sqrt{21}}{7}) = 2 \sqrt{21}$

Step 12.2.3

The final answer is $2 \sqrt{21}$ .

$y = 2 \sqrt{21}$

Step 13

Evaluate the second derivative at $t = - \frac{\sqrt{21}}{7}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{6}{{(- \frac{\sqrt{21}}{7})}^{3}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify the denominator.

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Step 14.1.1

Apply the product rule to $- \frac{\sqrt{21}}{7}$ .

$\frac{6}{{(- 1)}^{3} {(\frac{\sqrt{21}}{7})}^{3}}$

Step 14.1.2

Raise $- 1$ to the power of $3$ .

$\frac{6}{- {(\frac{\sqrt{21}}{7})}^{3}}$

Step 14.1.3

Apply the product rule to $\frac{\sqrt{21}}{7}$ .

$\frac{6}{- \frac{{\sqrt{21}}^{3}}{7^{3}}}$

Step 14.1.4

Simplify the numerator.

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Step 14.1.4.1

Rewrite ${\sqrt{21}}^{3}$ as $\sqrt{21^{3}}$ .

$\frac{6}{- \frac{\sqrt{21^{3}}}{7^{3}}}$

Step 14.1.4.2

Raise $21$ to the power of $3$ .

$\frac{6}{- \frac{\sqrt{9261}}{7^{3}}}$

Step 14.1.4.3

Rewrite $9261$ as $21^{2} \cdot 21$ .

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Step 14.1.4.3.1

Factor $441$ out of $9261$ .

$\frac{6}{- \frac{\sqrt{441 (21)}}{7^{3}}}$

Step 14.1.4.3.2

Rewrite $441$ as $21^{2}$ .

$\frac{6}{- \frac{\sqrt{21^{2} \cdot 21}}{7^{3}}}$

Step 14.1.4.4

Pull terms out from under the radical.

$\frac{6}{- \frac{21 \sqrt{21}}{7^{3}}}$

Step 14.1.5

Raise $7$ to the power of $3$ .

$\frac{6}{- \frac{21 \sqrt{21}}{343}}$

Step 14.1.6

Cancel the common factor of $21$ and $343$ .

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Step 14.1.6.1

Factor $7$ out of $21 \sqrt{21}$ .

$\frac{6}{- \frac{7 (3 \sqrt{21})}{343}}$

Step 14.1.6.2

Cancel the common factors.

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Step 14.1.6.2.1

Factor $7$ out of $343$ .

$\frac{6}{- \frac{7 (3 \sqrt{21})}{7 (49)}}$

Step 14.1.6.2.2

Cancel the common factor.

$\frac{6}{- \frac{7 (3 \sqrt{21})}{7 \cdot 49}}$

Step 14.1.6.2.3

Rewrite the expression.

$\frac{6}{- \frac{3 \sqrt{21}}{49}}$

Step 14.2

Multiply the numerator by the reciprocal of the denominator.

$6 (- \frac{49}{3 \sqrt{21}})$

Step 14.3

Cancel the common factor of $3$ .

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Step 14.3.1

Move the leading negative in $- \frac{49}{3 \sqrt{21}}$ into the numerator.

$6 \frac{- 49}{3 \sqrt{21}}$

Step 14.3.2

Factor $3$ out of $6$ .

$3 (2) \frac{- 49}{3 \sqrt{21}}$

Step 14.3.3

Factor $3$ out of $3 \sqrt{21}$ .

$3 (2) \frac{- 49}{3 (\sqrt{21})}$

Step 14.3.4

Cancel the common factor.

$3 \cdot 2 \frac{- 49}{3 \sqrt{21}}$

Step 14.3.5

Rewrite the expression.

$2 \frac{- 49}{\sqrt{21}}$

Step 14.4

Combine $2$ and $\frac{- 49}{\sqrt{21}}$ .

$\frac{2 \cdot - 49}{\sqrt{21}}$

Step 14.5

Simplify the expression.

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Step 14.5.1

Multiply $2$ by $- 49$ .

$\frac{- 98}{\sqrt{21}}$

Step 14.5.2

Move the negative in front of the fraction.

$- \frac{98}{\sqrt{21}}$

Step 14.6

Multiply $\frac{98}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$- (\frac{98}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}})$

Step 14.7

Combine and simplify the denominator.

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Step 14.7.1

Multiply $\frac{98}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$- \frac{98 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Step 14.7.2

Raise $\sqrt{21}$ to the power of $1$ .

$- \frac{98 \sqrt{21}}{{\sqrt{21}}^{1} \sqrt{21}}$

Step 14.7.3

Raise $\sqrt{21}$ to the power of $1$ .

$- \frac{98 \sqrt{21}}{{\sqrt{21}}^{1} {\sqrt{21}}^{1}}$

Step 14.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{98 \sqrt{21}}{{\sqrt{21}}^{1 + 1}}$

Step 14.7.5

Add $1$ and $1$ .

$- \frac{98 \sqrt{21}}{{\sqrt{21}}^{2}}$

Step 14.7.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 14.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$- \frac{98 \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}}$

Step 14.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{98 \sqrt{21}}{21^{\frac{1}{2} \cdot 2}}$

Step 14.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$- \frac{98 \sqrt{21}}{21^{\frac{2}{2}}}$

Step 14.7.6.4

Cancel the common factor of $2$ .

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Step 14.7.6.4.1

Cancel the common factor.

$- \frac{98 \sqrt{21}}{21^{\frac{2}{2}}}$

Step 14.7.6.4.2

Rewrite the expression.

$- \frac{98 \sqrt{21}}{21^{1}}$

Step 14.7.6.5

Evaluate the exponent.

$- \frac{98 \sqrt{21}}{21}$

Step 14.8

Cancel the common factor of $98$ and $21$ .

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Step 14.8.1

Factor $7$ out of $98 \sqrt{21}$ .

$- \frac{7 (14 \sqrt{21})}{21}$

Step 14.8.2

Cancel the common factors.

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Step 14.8.2.1

Factor $7$ out of $21$ .

$- \frac{7 (14 \sqrt{21})}{7 (3)}$

Step 14.8.2.2

Cancel the common factor.

$- \frac{7 (14 \sqrt{21})}{7 \cdot 3}$

Step 14.8.2.3

Rewrite the expression.

$- \frac{14 \sqrt{21}}{3}$

Step 15

$t = - \frac{\sqrt{21}}{7}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$t = - \frac{\sqrt{21}}{7}$ is a local maximum

Step 16

Find the y-value when $t = - \frac{\sqrt{21}}{7}$ .

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Step 16.1

Replace the variable $t$ with $- \frac{\sqrt{21}}{7}$ in the expression.

$f (- \frac{\sqrt{21}}{7}) = 7 (- \frac{\sqrt{21}}{7}) + \frac{3}{- \frac{\sqrt{21}}{7}}$

Step 16.2

Simplify the result.

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Step 16.2.1

Simplify each term.

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Step 16.2.1.1

Cancel the common factor of $7$ .

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Step 16.2.1.1.1

Move the leading negative in $- \frac{\sqrt{21}}{7}$ into the numerator.

$f (- \frac{\sqrt{21}}{7}) = 7 (\frac{- \sqrt{21}}{7}) + \frac{3}{- \frac{\sqrt{21}}{7}}$

Step 16.2.1.1.2

Cancel the common factor.

$f (- \frac{\sqrt{21}}{7}) = 7 (\frac{- \sqrt{21}}{7}) + \frac{3}{- \frac{\sqrt{21}}{7}}$

Step 16.2.1.1.3

Rewrite the expression.

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + \frac{3}{- \frac{\sqrt{21}}{7}}$

Step 16.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (- \frac{7}{\sqrt{21}})$

Step 16.2.1.3

Multiply $\frac{7}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (- (\frac{7}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}}))$

Step 16.2.1.4

Combine and simplify the denominator.

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Step 16.2.1.4.1

Multiply $\frac{7}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (- \frac{7 \sqrt{21}}{\sqrt{21} \sqrt{21}})$

Step 16.2.1.4.2

Raise $\sqrt{21}$ to the power of $1$ .

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (- \frac{7 \sqrt{21}}{\sqrt{21} \sqrt{21}})$

Step 16.2.1.4.3

Raise $\sqrt{21}$ to the power of $1$ .

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (- \frac{7 \sqrt{21}}{\sqrt{21} \sqrt{21}})$

Step 16.2.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (- \frac{7 \sqrt{21}}{{\sqrt{21}}^{1 + 1}})$

Step 16.2.1.4.5

Add $1$ and $1$ .

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (- \frac{7 \sqrt{21}}{{\sqrt{21}}^{2}})$

Step 16.2.1.4.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 16.2.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (- \frac{7 \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}})$

Step 16.2.1.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (- \frac{7 \sqrt{21}}{21^{\frac{1}{2} \cdot 2}})$

Step 16.2.1.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (- \frac{7 \sqrt{21}}{21^{\frac{2}{2}}})$

Step 16.2.1.4.6.4

Cancel the common factor of $2$ .

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Step 16.2.1.4.6.4.1

Cancel the common factor.

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (- \frac{7 \sqrt{21}}{21^{\frac{2}{2}}})$

Step 16.2.1.4.6.4.2

Rewrite the expression.

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (- \frac{7 \sqrt{21}}{21})$

Step 16.2.1.4.6.5

Evaluate the exponent.

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (- \frac{7 \sqrt{21}}{21})$

Step 16.2.1.5

Cancel the common factor of $3$ .

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Step 16.2.1.5.1

Move the leading negative in $- \frac{7 \sqrt{21}}{21}$ into the numerator.

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (\frac{- 7 \sqrt{21}}{21})$

Step 16.2.1.5.2

Factor $3$ out of $21$ .

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (\frac{- 7 \sqrt{21}}{3 (7)})$

Step 16.2.1.5.3

Cancel the common factor.

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + 3 (\frac{- 7 \sqrt{21}}{3 \cdot 7})$

Step 16.2.1.5.4

Rewrite the expression.

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + \frac{- 7 \sqrt{21}}{7}$

Step 16.2.1.6

Cancel the common factor of $- 7$ and $7$ .

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Step 16.2.1.6.1

Factor $7$ out of $- 7 \sqrt{21}$ .

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + \frac{7 (- \sqrt{21})}{7}$

Step 16.2.1.6.2

Cancel the common factors.

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Step 16.2.1.6.2.1

Factor $7$ out of $7$ .

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + \frac{7 (- \sqrt{21})}{7 (1)}$

Step 16.2.1.6.2.2

Cancel the common factor.

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + \frac{7 (- \sqrt{21})}{7 \cdot 1}$

Step 16.2.1.6.2.3

Rewrite the expression.

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} + \frac{- \sqrt{21}}{1}$

Step 16.2.1.6.2.4

Divide $- \sqrt{21}$ by $1$ .

$f (- \frac{\sqrt{21}}{7}) = - \sqrt{21} - \sqrt{21}$

Step 16.2.2

Subtract $\sqrt{21}$ from $- \sqrt{21}$ .

$f (- \frac{\sqrt{21}}{7}) = - 2 \sqrt{21}$

Step 16.2.3

The final answer is $- 2 \sqrt{21}$ .

$y = - 2 \sqrt{21}$

Step 17

These are the local extrema for $f (t) = 7 t + \frac{3}{t}$ .

$(\frac{\sqrt{21}}{7}, 2 \sqrt{21})$ is a local minima

$(- \frac{\sqrt{21}}{7}, - 2 \sqrt{21})$ is a local maxima

Step 18