Calculus Examples

$f (x) = a x - 3 x \ln (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $a x - 3 x \ln (x)$ with respect to $x$ is $\frac{d}{d x} [a x] + \frac{d}{d x} [- 3 x \ln (x)]$ .

$\frac{d}{d x} [a x] + \frac{d}{d x} [- 3 x \ln (x)]$

Step 1.2

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

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Step 1.2.1

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

$a \frac{d}{d x} [x] + \frac{d}{d x} [- 3 x \ln (x)]$

Step 1.2.2

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

$a \cdot 1 + \frac{d}{d x} [- 3 x \ln (x)]$

Step 1.2.3

Multiply $a$ by $1$ .

$a + \frac{d}{d x} [- 3 x \ln (x)]$

Step 1.3

Evaluate $\frac{d}{d x} [- 3 x \ln (x)]$ .

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Step 1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x \ln (x)$ with respect to $x$ is $- 3 \frac{d}{d x} [x \ln (x)]$ .

$a - 3 \frac{d}{d x} [x \ln (x)]$

Step 1.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \ln (x)$ .

$a - 3 (x \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x])$

Step 1.3.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$a - 3 (x \frac{1}{x} + \ln (x) \frac{d}{d x} [x])$

Step 1.3.4

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

$a - 3 (x \frac{1}{x} + \ln (x) \cdot 1)$

Step 1.3.5

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

$a - 3 (\frac{x}{x} + \ln (x) \cdot 1)$

Step 1.3.6

Cancel the common factor of $x$ .

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Step 1.3.6.1

Cancel the common factor.

$a - 3 (\frac{x}{x} + \ln (x) \cdot 1)$

Step 1.3.6.2

Rewrite the expression.

$a - 3 (1 + \ln (x) \cdot 1)$

Step 1.3.7

Multiply $\ln (x)$ by $1$ .

$a - 3 (1 + \ln (x))$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$a - 3 \cdot 1 - 3 \ln (x)$

Step 1.4.2

Multiply $- 3$ by $1$ .

$a - 3 - 3 \ln (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $a - 3 - 3 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [a] + \frac{d}{d x} [- 3] + \frac{d}{d x} [- 3 \ln (x)]$ .

$f'' (x) = \frac{d}{d x} (a) + \frac{d}{d x} (- 3) + \frac{d}{d x} (- 3 \ln (x))$

Step 2.1.2

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

$f'' (x) = 0 + \frac{d}{d x} (- 3) + \frac{d}{d x} (- 3 \ln (x))$

Step 2.1.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$f'' (x) = 0 + 0 + \frac{d}{d x} (- 3 \ln (x))$

Step 2.2

Evaluate $\frac{d}{d x} [- 3 \ln (x)]$ .

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Step 2.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 \ln (x)$ with respect to $x$ is $- 3 \frac{d}{d x} [\ln (x)]$ .

$f'' (x) = 0 + 0 - 3 \frac{d}{d x} \ln (x)$

Step 2.2.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

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

Step 2.2.3

Combine $- 3$ and $\frac{1}{x}$ .

$f'' (x) = 0 + 0 + \frac{- 3}{x}$

Step 2.2.4

Move the negative in front of the fraction.

$f'' (x) = 0 + 0 - \frac{3}{x}$

Step 2.3

Combine terms.

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Step 2.3.1

Add $0$ and $0$ .

$f'' (x) = 0 - \frac{3}{x}$

Step 2.3.2

Subtract $\frac{3}{x}$ from $0$ .

$f'' (x) = - \frac{3}{x}$

Step 3

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

$a - 3 - 3 \ln (x) = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

By the Sum Rule, the derivative of $a x - 3 x \ln (x)$ with respect to $x$ is $\frac{d}{d x} [a x] + \frac{d}{d x} [- 3 x \ln (x)]$ .

$\frac{d}{d x} [a x] + \frac{d}{d x} [- 3 x \ln (x)]$

Step 4.1.2

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

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Step 4.1.2.1

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

$a \frac{d}{d x} [x] + \frac{d}{d x} [- 3 x \ln (x)]$

Step 4.1.2.2

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

$a \cdot 1 + \frac{d}{d x} [- 3 x \ln (x)]$

Step 4.1.2.3

Multiply $a$ by $1$ .

$a + \frac{d}{d x} [- 3 x \ln (x)]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- 3 x \ln (x)]$ .

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Step 4.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x \ln (x)$ with respect to $x$ is $- 3 \frac{d}{d x} [x \ln (x)]$ .

$a - 3 \frac{d}{d x} [x \ln (x)]$

Step 4.1.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \ln (x)$ .

$a - 3 (x \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x])$

Step 4.1.3.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$a - 3 (x \frac{1}{x} + \ln (x) \frac{d}{d x} [x])$

Step 4.1.3.4

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

$a - 3 (x \frac{1}{x} + \ln (x) \cdot 1)$

Step 4.1.3.5

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

$a - 3 (\frac{x}{x} + \ln (x) \cdot 1)$

Step 4.1.3.6

Cancel the common factor of $x$ .

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Step 4.1.3.6.1

Cancel the common factor.

$a - 3 (\frac{x}{x} + \ln (x) \cdot 1)$

Step 4.1.3.6.2

Rewrite the expression.

$a - 3 (1 + \ln (x) \cdot 1)$

Step 4.1.3.7

Multiply $\ln (x)$ by $1$ .

$a - 3 (1 + \ln (x))$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Apply the distributive property.

$a - 3 \cdot 1 - 3 \ln (x)$

Step 4.1.4.2

Multiply $- 3$ by $1$ .

$f' (x) = a - 3 - 3 \ln (x)$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $a - 3 - 3 \ln (x)$ .

$a - 3 - 3 \ln (x)$

Step 5

Set the first derivative equal to $0$ then solve the equation $a - 3 - 3 \ln (x) = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$a - 3 - 3 \ln (x) = 0$

Step 5.2

Move all terms not containing $\ln (x)$ to the right side of the equation.

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Step 5.2.1

Subtract $a$ from both sides of the equation.

$- 3 - 3 \ln (x) = - a$

Step 5.2.2

Add $3$ to both sides of the equation.

$- 3 \ln (x) = - a + 3$

Step 5.3

Divide each term in $- 3 \ln (x) = - a + 3$ by $- 3$ and simplify.

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Step 5.3.1

Divide each term in $- 3 \ln (x) = - a + 3$ by $- 3$ .

$\frac{- 3 \ln (x)}{- 3} = \frac{- a}{- 3} + \frac{3}{- 3}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 3 \ln (x)}{- 3} = \frac{- a}{- 3} + \frac{3}{- 3}$

Step 5.3.2.1.2

Divide $\ln (x)$ by $1$ .

$\ln (x) = \frac{- a}{- 3} + \frac{3}{- 3}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Dividing two negative values results in a positive value.

$\ln (x) = \frac{a}{3} + \frac{3}{- 3}$

Step 5.3.3.1.2

Divide $3$ by $- 3$ .

$\ln (x) = \frac{a}{3} - 1$

Step 5.4

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x)} = e^{\frac{a}{3} - 1}$

Step 5.5

Rewrite $\ln (x) = \frac{a}{3} - 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{\frac{a}{3} - 1} = x$

Step 5.6

Rewrite the equation as $x = e^{\frac{a}{3} - 1}$ .

$x = e^{\frac{a}{3} - 1}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the argument in $\ln (x)$ less than or equal to $0$ to find where the expression is undefined.

$x \leq 0$

Step 6.2

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 7

Critical points to evaluate.

$x = e^{\frac{a}{3} - 1}$

Step 8

Evaluate the second derivative at $x = e^{\frac{a}{3} - 1}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{3}{e^{\frac{a}{3} - 1}}$

Step 9

Simplify the denominator.

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Step 9.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{3}{e^{\frac{a}{3} - 1 \cdot \frac{3}{3}}}$

Step 9.2

Combine $- 1$ and $\frac{3}{3}$ .

$- \frac{3}{e^{\frac{a}{3} + \frac{- 1 \cdot 3}{3}}}$

Step 9.3

Combine the numerators over the common denominator.

$- \frac{3}{e^{\frac{a - 1 \cdot 3}{3}}}$

Step 9.4

Multiply $- 1$ by $3$ .

$- \frac{3}{e^{\frac{a - 3}{3}}}$

Step 10

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 11