Calculus Examples

$y = x + \frac{1}{4 x}$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

Differentiate.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $x + \frac{1}{4 x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{4 x}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{4 x}]$

Step 1.1.2

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

$1 + \frac{d}{d x} [\frac{1}{4 x}]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{1}{4 x}]$ .

Tap for more steps...

Step 1.2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4 x}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [\frac{1}{x}]$ .

$1 + \frac{1}{4} \frac{d}{d x} [\frac{1}{x}]$

Step 1.2.2

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

$1 + \frac{1}{4} \frac{d}{d x} [x^{- 1}]$

Step 1.2.3

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

$1 + \frac{1}{4} (- x^{- 2})$

Step 1.2.4

Combine $\frac{1}{4}$ and $x^{- 2}$ .

$1 - \frac{x^{- 2}}{4}$

Step 1.2.5

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 - \frac{1}{4 x^{2}}$

Step 1.3

Reorder terms.

$- \frac{1}{4 x^{2}} + 1$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $- \frac{1}{4 x^{2}} + 1$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{4 x^{2}}] + \frac{d}{d x} [1]$ .

$f'' (x) = \frac{d}{d x} (- \frac{1}{4 x^{2}}) + \frac{d}{d x} (1)$

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{1}{4 x^{2}}]$ .

Tap for more steps...

Step 2.2.1

Since $- \frac{1}{4}$ is constant with respect to $x$ , the derivative of $- \frac{1}{4 x^{2}}$ with respect to $x$ is $- \frac{1}{4} \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$f'' (x) = - \frac{1}{4} \cdot \frac{d}{d x} \frac{1}{x^{2}} + \frac{d}{d x} (1)$

Step 2.2.2

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

$f'' (x) = - \frac{1}{4} \cdot \frac{d}{d x} {(x^{2})}^{- 1} + \frac{d}{d x} (1)$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

Tap for more steps...

Step 2.2.3.1

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

$f'' (x) = - \frac{1}{4} \cdot (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{2})) + \frac{d}{d x} (1)$

Step 2.2.3.2

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

$f'' (x) = - \frac{1}{4} \cdot (- u^{- 2} \frac{d}{d x} x^{2}) + \frac{d}{d x} (1)$

Step 2.2.3.3

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

$f'' (x) = - \frac{1}{4} \cdot (- {(x^{2})}^{- 2} \frac{d}{d x} x^{2}) + \frac{d}{d x} (1)$

Step 2.2.4

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

$f'' (x) = - \frac{1}{4} \cdot (- {(x^{2})}^{- 2} (2 x)) + \frac{d}{d x} (1)$

Step 2.2.5

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

Tap for more steps...

Step 2.2.5.1

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

$f'' (x) = - \frac{1}{4} \cdot (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} (1)$

Step 2.2.5.2

Multiply $2$ by $- 2$ .

$f'' (x) = - \frac{1}{4} \cdot (- x^{- 4} (2 x)) + \frac{d}{d x} (1)$

Step 2.2.6

Multiply $2$ by $- 1$ .

$f'' (x) = - \frac{1}{4} \cdot (- 2 x^{- 4} x) + \frac{d}{d x} (1)$

Step 2.2.7

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

$f'' (x) = - \frac{1}{4} \cdot (- 2 (x \cdot x^{- 4})) + \frac{d}{d x} (1)$

Step 2.2.8

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

$f'' (x) = - \frac{1}{4} \cdot (- 2 x^{1 - 4}) + \frac{d}{d x} (1)$

Step 2.2.9

Subtract $4$ from $1$ .

$f'' (x) = - \frac{1}{4} \cdot (- 2 x^{- 3}) + \frac{d}{d x} (1)$

Step 2.2.10

Multiply $- 2$ by $- 1$ .

$f'' (x) = 2 (\frac{1}{4}) \cdot x^{- 3} + \frac{d}{d x} (1)$

Step 2.2.11

Combine $2$ and $\frac{1}{4}$ .

$f'' (x) = \frac{2}{4} x^{- 3} + \frac{d}{d x} (1)$

Step 2.2.12

Combine $\frac{2}{4}$ and $x^{- 3}$ .

$f'' (x) = \frac{2 x^{- 3}}{4} + \frac{d}{d x} (1)$

Step 2.2.13

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{2}{4 x^{3}} + \frac{d}{d x} (1)$

Step 2.2.14

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 2.2.14.1

Factor $2$ out of $2$ .

$f'' (x) = \frac{2 (1)}{4 x^{3}} + \frac{d}{d x} (1)$

Step 2.2.14.2

Cancel the common factors.

Tap for more steps...

Step 2.2.14.2.1

Factor $2$ out of $4 x^{3}$ .

$f'' (x) = \frac{2 (1)}{2 (2 x^{3})} + \frac{d}{d x} (1)$

Step 2.2.14.2.2

Cancel the common factor.

$f'' (x) = \frac{2 \cdot 1}{2 (2 x^{3})} + \frac{d}{d x} (1)$

Step 2.2.14.2.3

Rewrite the expression.

$f'' (x) = \frac{1}{2 x^{3}} + \frac{d}{d x} (1)$

Step 2.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.3.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = \frac{1}{2 x^{3}} + 0$

Step 2.3.2

Add $\frac{1}{2 x^{3}}$ and $0$ .

$f'' (x) = \frac{1}{2 x^{3}}$

Step 3

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

$- \frac{1}{4 x^{2}} + 1 = 0$

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

Differentiate.

Tap for more steps...

Step 4.1.1.1

By the Sum Rule, the derivative of $x + \frac{1}{4 x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{4 x}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{4 x}]$

Step 4.1.1.2

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

$1 + \frac{d}{d x} [\frac{1}{4 x}]$

Step 4.1.2

Evaluate $\frac{d}{d x} [\frac{1}{4 x}]$ .

Tap for more steps...

Step 4.1.2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4 x}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [\frac{1}{x}]$ .

$1 + \frac{1}{4} \frac{d}{d x} [\frac{1}{x}]$

Step 4.1.2.2

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

$1 + \frac{1}{4} \frac{d}{d x} [x^{- 1}]$

Step 4.1.2.3

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

$1 + \frac{1}{4} (- x^{- 2})$

Step 4.1.2.4

Combine $\frac{1}{4}$ and $x^{- 2}$ .

$1 - \frac{x^{- 2}}{4}$

Step 4.1.2.5

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 - \frac{1}{4 x^{2}}$

Step 4.1.3

Reorder terms.

$f' (x) = - \frac{1}{4 x^{2}} + 1$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{1}{4 x^{2}} + 1$ .

$- \frac{1}{4 x^{2}} + 1$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{1}{4 x^{2}} + 1 = 0$ .

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

$- \frac{1}{4 x^{2}} + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$- \frac{1}{4 x^{2}} = - 1$

Step 5.3

Find the LCD of the terms in the equation.

Tap for more steps...

Step 5.3.1

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

$4 x^{2}, 1$

Step 5.3.2

The LCM of one and any expression is the expression.

$4 x^{2}$

Step 5.4

Multiply each term in $- \frac{1}{4 x^{2}} = - 1$ by $4 x^{2}$ to eliminate the fractions.

Tap for more steps...

Step 5.4.1

Multiply each term in $- \frac{1}{4 x^{2}} = - 1$ by $4 x^{2}$ .

$- \frac{1}{4 x^{2}} (4 x^{2}) = - (4 x^{2})$

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $4 x^{2}$ .

Tap for more steps...

Step 5.4.2.1.1

Move the leading negative in $- \frac{1}{4 x^{2}}$ into the numerator.

$\frac{- 1}{4 x^{2}} (4 x^{2}) = - (4 x^{2})$

Step 5.4.2.1.2

Cancel the common factor.

$\frac{- 1}{4 x^{2}} (4 x^{2}) = - (4 x^{2})$

Step 5.4.2.1.3

Rewrite the expression.

$- 1 = - (4 x^{2})$

Step 5.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.1

Multiply $4$ by $- 1$ .

$- 1 = - 4 x^{2}$

Step 5.5

Solve the equation.

Tap for more steps...

Step 5.5.1

Rewrite the equation as $- 4 x^{2} = - 1$ .

$- 4 x^{2} = - 1$

Step 5.5.2

Divide each term in $- 4 x^{2} = - 1$ by $- 4$ and simplify.

Tap for more steps...

Step 5.5.2.1

Divide each term in $- 4 x^{2} = - 1$ by $- 4$ .

$\frac{- 4 x^{2}}{- 4} = \frac{- 1}{- 4}$

Step 5.5.2.2

Simplify the left side.

Tap for more steps...

Step 5.5.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 5.5.2.2.1.1

Cancel the common factor.

$\frac{- 4 x^{2}}{- 4} = \frac{- 1}{- 4}$

Step 5.5.2.2.1.2

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

$x^{2} = \frac{- 1}{- 4}$

Step 5.5.2.3

Simplify the right side.

Tap for more steps...

Step 5.5.2.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{1}{4}$

Step 5.5.3

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

$x = \pm \sqrt{\frac{1}{4}}$

Step 5.5.4

Simplify $\pm \sqrt{\frac{1}{4}}$ .

Tap for more steps...

Step 5.5.4.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{4}}$

Step 5.5.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{4}}$

Step 5.5.4.3

Simplify the denominator.

Tap for more steps...

Step 5.5.4.3.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \frac{1}{\sqrt{2^{2}}}$

Step 5.5.4.3.2

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

$x = \pm \frac{1}{2}$

Step 5.5.5

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

Tap for more steps...

Step 5.5.5.1

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

$x = \frac{1}{2}$

Step 5.5.5.2

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

$x = - \frac{1}{2}$

Step 5.5.5.3

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

$x = \frac{1}{2}, - \frac{1}{2}$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

Set the denominator in $\frac{1}{4 x^{2}}$ equal to $0$ to find where the expression is undefined.

$4 x^{2} = 0$

Step 6.2

Solve for $x$ .

Tap for more steps...

Step 6.2.1

Divide each term in $4 x^{2} = 0$ by $4$ and simplify.

Tap for more steps...

Step 6.2.1.1

Divide each term in $4 x^{2} = 0$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{0}{4}$

Step 6.2.1.2

Simplify the left side.

Tap for more steps...

Step 6.2.1.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.2.1.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{0}{4}$

Step 6.2.1.2.1.2

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

$x^{2} = \frac{0}{4}$

Step 6.2.1.3

Simplify the right side.

Tap for more steps...

Step 6.2.1.3.1

Divide $0$ by $4$ .

$x^{2} = 0$

Step 6.2.2

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

$x = \pm \sqrt{0}$

Step 6.2.3

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 6.2.3.1

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

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

Step 6.2.3.2

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

$x = \pm 0$

Step 6.2.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 7

Critical points to evaluate.

$x = \frac{1}{2}, - \frac{1}{2}$

Step 8

Evaluate the second derivative at $x = \frac{1}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{1}{2 {(\frac{1}{2})}^{3}}$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify the denominator.

Tap for more steps...

Step 9.1.1

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

$\frac{1}{2 {(2^{- 1})}^{3}}$

Step 9.1.2

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

$\frac{1}{2 {({(2^{1})}^{- 1})}^{3}}$

Step 9.1.3

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

$\frac{1}{2 {(2^{1 \cdot - 1})}^{3}}$

Step 9.1.4

Multiply $- 1$ by $1$ .

$\frac{1}{2 {(2^{- 1})}^{3}}$

Step 9.1.5

Multiply the exponents in ${(2^{- 1})}^{3}$ .

Tap for more steps...

Step 9.1.5.1

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

$\frac{1}{2 \cdot 2^{- 1 \cdot 3}}$

Step 9.1.5.2

Multiply $- 1$ by $3$ .

$\frac{1}{2 \cdot 2^{- 3}}$

Step 9.1.6

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

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

Step 9.1.7

Subtract $3$ from $1$ .

$\frac{1}{2^{- 2}}$

Step 9.2

Simplify the denominator.

Tap for more steps...

Step 9.2.1

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

$\frac{1}{\frac{1}{2^{2}}}$

Step 9.2.2

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

$\frac{1}{\frac{1}{4}}$

Step 9.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 4$

Step 9.4

Multiply $4$ by $1$ .

$4$

Step 10

$x = \frac{1}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{1}{2}$ is a local minimum

Step 11

Find the y-value when $x = \frac{1}{2}$ .

Tap for more steps...

Step 11.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

$f (\frac{1}{2}) = (\frac{1}{2}) + \frac{1}{4 (\frac{1}{2})}$

Step 11.2

Simplify the result.

Tap for more steps...

Step 11.2.1

Simplify each term.

Tap for more steps...

Step 11.2.1.1

Combine $4$ and $\frac{1}{2}$ .

$f (\frac{1}{2}) = \frac{1}{2} + \frac{1}{\frac{4}{2}}$

Step 11.2.1.2

Divide $4$ by $2$ .

$f (\frac{1}{2}) = \frac{1}{2} + \frac{1}{2}$

Step 11.2.2

Combine fractions.

Tap for more steps...

Step 11.2.2.1

Combine the numerators over the common denominator.

$f (\frac{1}{2}) = \frac{1 + 1}{2}$

Step 11.2.2.2

Simplify the expression.

Tap for more steps...

Step 11.2.2.2.1

Add $1$ and $1$ .

$f (\frac{1}{2}) = \frac{2}{2}$

Step 11.2.2.2.2

Divide $2$ by $2$ .

$f (\frac{1}{2}) = 1$

Step 11.2.3

The final answer is $1$ .

$y = 1$

Step 12

Evaluate the second derivative at $x = - \frac{1}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{1}{2 {(- \frac{1}{2})}^{3}}$

Step 13

Evaluate the second derivative.

Tap for more steps...

Step 13.1

Simplify the denominator.

Tap for more steps...

Step 13.1.1

Apply the product rule to $- \frac{1}{2}$ .

$\frac{1}{2 {(- 1)}^{3} {(\frac{1}{2})}^{3}}$

Step 13.1.2

Combine exponents.

Tap for more steps...

Step 13.1.2.1

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

$\frac{1}{{(2^{- 1})}^{3} \cdot 2 {(- 1)}^{3}}$

Step 13.1.2.2

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

$\frac{1}{{({(2^{1})}^{- 1})}^{3} \cdot 2 {(- 1)}^{3}}$

Step 13.1.2.3

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

$\frac{1}{{(2^{1 \cdot - 1})}^{3} \cdot 2 {(- 1)}^{3}}$

Step 13.1.2.4

Multiply $- 1$ by $1$ .

$\frac{1}{{(2^{- 1})}^{3} \cdot 2 {(- 1)}^{3}}$

Step 13.1.2.5

Multiply the exponents in ${(2^{- 1})}^{3}$ .

Tap for more steps...

Step 13.1.2.5.1

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

$\frac{1}{2^{- 1 \cdot 3} \cdot 2 {(- 1)}^{3}}$

Step 13.1.2.5.2

Multiply $- 1$ by $3$ .

$\frac{1}{2^{- 3} \cdot 2 {(- 1)}^{3}}$

Step 13.1.2.6

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

$\frac{1}{2^{- 3 + 1} {(- 1)}^{3}}$

Step 13.1.2.7

Add $- 3$ and $1$ .

$\frac{1}{2^{- 2} {(- 1)}^{3}}$

Step 13.1.3

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

$\frac{1}{\frac{1}{2^{2}} {(- 1)}^{3}}$

Step 13.1.4

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

$\frac{1}{\frac{1}{4} {(- 1)}^{3}}$

Step 13.1.5

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

$\frac{1}{\frac{1}{4} \cdot - 1}$

Step 13.2

Simplify terms.

Tap for more steps...

Step 13.2.1

Combine $\frac{1}{4}$ and $- 1$ .

$\frac{1}{\frac{- 1}{4}}$

Step 13.2.2

Move the negative in front of the fraction.

$\frac{1}{- \frac{1}{4}}$

Step 13.2.3

Cancel the common factor of $1$ and $- 1$ .

Tap for more steps...

Step 13.2.3.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- \frac{1}{4}}$

Step 13.2.3.2

Move the negative in front of the fraction.

$- \frac{1}{\frac{1}{4}}$

Step 13.3

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 4)$

Step 13.4

Multiply $- (1 \cdot 4)$ .

Tap for more steps...

Step 13.4.1

Multiply $4$ by $1$ .

$- 1 \cdot 4$

Step 13.4.2

Multiply $- 1$ by $4$ .

$- 4$

Step 14

$x = - \frac{1}{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \frac{1}{2}$ is a local maximum

Step 15

Find the y-value when $x = - \frac{1}{2}$ .

Tap for more steps...

Step 15.1

Replace the variable $x$ with $- \frac{1}{2}$ in the expression.

$f (- \frac{1}{2}) = (- \frac{1}{2}) + \frac{1}{4 (- \frac{1}{2})}$

Step 15.2

Simplify the result.

Tap for more steps...

Step 15.2.1

Simplify each term.

Tap for more steps...

Step 15.2.1.1

Cancel the common factor of $1$ and $- 1$ .

Tap for more steps...

Step 15.2.1.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$f (- \frac{1}{2}) = - \frac{1}{2} + \frac{- 1 \cdot - 1}{4 (- \frac{1}{2})}$

Step 15.2.1.1.2

Move the negative in front of the fraction.

$f (- \frac{1}{2}) = - \frac{1}{2} - \frac{1}{4 (\frac{1}{2})}$

Step 15.2.1.2

Combine $4$ and $\frac{1}{2}$ .

$f (- \frac{1}{2}) = - \frac{1}{2} - \frac{1}{\frac{4}{2}}$

Step 15.2.1.3

Divide $4$ by $2$ .

$f (- \frac{1}{2}) = - \frac{1}{2} - \frac{1}{2}$

Step 15.2.2

Combine fractions.

Tap for more steps...

Step 15.2.2.1

Combine the numerators over the common denominator.

$f (- \frac{1}{2}) = \frac{- 1 - 1}{2}$

Step 15.2.2.2

Simplify the expression.

Tap for more steps...

Step 15.2.2.2.1

Subtract $1$ from $- 1$ .

$f (- \frac{1}{2}) = \frac{- 2}{2}$

Step 15.2.2.2.2

Divide $- 2$ by $2$ .

$f (- \frac{1}{2}) = - 1$

Step 15.2.3

The final answer is $- 1$ .

$y = - 1$

Step 16

These are the local extrema for $f (x) = x + \frac{1}{4 x}$ .

$(\frac{1}{2}, 1)$ is a local minima

$(- \frac{1}{2}, - 1)$ is a local maxima

Step 17