Calculus Examples

$f (x) = x^{3} + x^{2} - 5 x + 8$ ; $(0, \infty)$

Step 1

Find the critical points.

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Step 1.1

Find the first 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 $x^{3} + x^{2} - 5 x + 8$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [8]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [8]$

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 $n = 3$ .

$3 x^{2} + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [8]$

Step 1.1.1.1.3

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

$3 x^{2} + 2 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [8]$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

$3 x^{2} + 2 x - 5 \frac{d}{d x} [x] + \frac{d}{d x} [8]$

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 $n = 1$ .

$3 x^{2} + 2 x - 5 \cdot 1 + \frac{d}{d x} [8]$

Step 1.1.1.2.3

Multiply $- 5$ by $1$ .

$3 x^{2} + 2 x - 5 + \frac{d}{d x} [8]$

Step 1.1.1.3

Differentiate using the Constant Rule.

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Step 1.1.1.3.1

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

$3 x^{2} + 2 x - 5 + 0$

Step 1.1.1.3.2

Add $3 x^{2} + 2 x - 5$ and $0$ .

$f' (x) = 3 x^{2} + 2 x - 5$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} + 2 x - 5$ .

$3 x^{2} + 2 x - 5$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} + 2 x - 5 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$3 x^{2} + 2 x - 5 = 0$

Step 1.2.2

Factor by grouping.

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Step 1.2.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 5 = - 15$ and whose sum is $b = 2$ .

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Step 1.2.2.1.1

Factor $2$ out of $2 x$ .

$3 x^{2} + 2 (x) - 5 = 0$

Step 1.2.2.1.2

Rewrite $2$ as $- 3$ plus $5$

$3 x^{2} + (- 3 + 5) x - 5 = 0$

Step 1.2.2.1.3

Apply the distributive property.

$3 x^{2} - 3 x + 5 x - 5 = 0$

Step 1.2.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.2.1

Group the first two terms and the last two terms.

$(3 x^{2} - 3 x) + 5 x - 5 = 0$

Step 1.2.2.2.2

Factor out the greatest common factor (GCF) from each group.

$3 x (x - 1) + 5 (x - 1) = 0$

Step 1.2.2.3

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$(x - 1) (3 x + 5) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$3 x + 5 = 0$

Step 1.2.4

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.5

Set $3 x + 5$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $3 x + 5$ equal to $0$ .

$3 x + 5 = 0$

Step 1.2.5.2

Solve $3 x + 5 = 0$ for $x$ .

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Step 1.2.5.2.1

Subtract $5$ from both sides of the equation.

$3 x = - 5$

Step 1.2.5.2.2

Divide each term in $3 x = - 5$ by $3$ and simplify.

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Step 1.2.5.2.2.1

Divide each term in $3 x = - 5$ by $3$ .

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 1.2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{3}$

Step 1.2.5.2.2.3

Simplify the right side.

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Step 1.2.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{3}$

Step 1.2.6

The final solution is all the values that make $(x - 1) (3 x + 5) = 0$ true.

$x = 1, - \frac{5}{3}$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

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 1.4

Evaluate $x^{3} + x^{2} - 5 x + 8$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 1$ .

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Step 1.4.1.1

Substitute $1$ for $x$ .

${(1)}^{3} + {(1)}^{2} - 5 \cdot 1 + 8$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

One to any power is one.

$1 + {(1)}^{2} - 5 \cdot 1 + 8$

Step 1.4.1.2.1.2

One to any power is one.

$1 + 1 - 5 \cdot 1 + 8$

Step 1.4.1.2.1.3

Multiply $- 5$ by $1$ .

$1 + 1 - 5 + 8$

Step 1.4.1.2.2

Simplify by adding and subtracting.

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Step 1.4.1.2.2.1

Add $1$ and $1$ .

$2 - 5 + 8$

Step 1.4.1.2.2.2

Subtract $5$ from $2$ .

$- 3 + 8$

Step 1.4.1.2.2.3

Add $- 3$ and $8$ .

$5$

Step 1.4.2

Evaluate at $x = - \frac{5}{3}$ .

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Step 1.4.2.1

Substitute $- \frac{5}{3}$ for $x$ .

${(- \frac{5}{3})}^{3} + {(- \frac{5}{3})}^{2} - 5 (- \frac{5}{3}) + 8$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify each term.

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Step 1.4.2.2.1.1

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

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Step 1.4.2.2.1.1.1

Apply the product rule to $- \frac{5}{3}$ .

${(- 1)}^{3} {(\frac{5}{3})}^{3} + {(- \frac{5}{3})}^{2} - 5 (- \frac{5}{3}) + 8$

Step 1.4.2.2.1.1.2

Apply the product rule to $\frac{5}{3}$ .

${(- 1)}^{3} \frac{5^{3}}{3^{3}} + {(- \frac{5}{3})}^{2} - 5 (- \frac{5}{3}) + 8$

Step 1.4.2.2.1.2

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

$- \frac{5^{3}}{3^{3}} + {(- \frac{5}{3})}^{2} - 5 (- \frac{5}{3}) + 8$

Step 1.4.2.2.1.3

Raise $5$ to the power of $3$ .

$- \frac{125}{3^{3}} + {(- \frac{5}{3})}^{2} - 5 (- \frac{5}{3}) + 8$

Step 1.4.2.2.1.4

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

$- \frac{125}{27} + {(- \frac{5}{3})}^{2} - 5 (- \frac{5}{3}) + 8$

Step 1.4.2.2.1.5

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

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Step 1.4.2.2.1.5.1

Apply the product rule to $- \frac{5}{3}$ .

$- \frac{125}{27} + {(- 1)}^{2} {(\frac{5}{3})}^{2} - 5 (- \frac{5}{3}) + 8$

Step 1.4.2.2.1.5.2

Apply the product rule to $\frac{5}{3}$ .

$- \frac{125}{27} + {(- 1)}^{2} \frac{5^{2}}{3^{2}} - 5 (- \frac{5}{3}) + 8$

Step 1.4.2.2.1.6

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

$- \frac{125}{27} + 1 \frac{5^{2}}{3^{2}} - 5 (- \frac{5}{3}) + 8$

Step 1.4.2.2.1.7

Multiply $\frac{5^{2}}{3^{2}}$ by $1$ .

$- \frac{125}{27} + \frac{5^{2}}{3^{2}} - 5 (- \frac{5}{3}) + 8$

Step 1.4.2.2.1.8

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

$- \frac{125}{27} + \frac{25}{3^{2}} - 5 (- \frac{5}{3}) + 8$

Step 1.4.2.2.1.9

Raise $3$ to the power of $2$ .

$- \frac{125}{27} + \frac{25}{9} - 5 (- \frac{5}{3}) + 8$

Step 1.4.2.2.1.10

Multiply $- 5 (- \frac{5}{3})$ .

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Step 1.4.2.2.1.10.1

Multiply $- 1$ by $- 5$ .

$- \frac{125}{27} + \frac{25}{9} + 5 (\frac{5}{3}) + 8$

Step 1.4.2.2.1.10.2

Combine $5$ and $\frac{5}{3}$ .

$- \frac{125}{27} + \frac{25}{9} + \frac{5 \cdot 5}{3} + 8$

Step 1.4.2.2.1.10.3

Multiply $5$ by $5$ .

$- \frac{125}{27} + \frac{25}{9} + \frac{25}{3} + 8$

Step 1.4.2.2.2

Find the common denominator.

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Step 1.4.2.2.2.1

Multiply $\frac{25}{9}$ by $\frac{3}{3}$ .

$- \frac{125}{27} + \frac{25}{9} \cdot \frac{3}{3} + \frac{25}{3} + 8$

Step 1.4.2.2.2.2

Multiply $\frac{25}{9}$ by $\frac{3}{3}$ .

$- \frac{125}{27} + \frac{25 \cdot 3}{9 \cdot 3} + \frac{25}{3} + 8$

Step 1.4.2.2.2.3

Multiply $\frac{25}{3}$ by $\frac{9}{9}$ .

$- \frac{125}{27} + \frac{25 \cdot 3}{9 \cdot 3} + \frac{25}{3} \cdot \frac{9}{9} + 8$

Step 1.4.2.2.2.4

Multiply $\frac{25}{3}$ by $\frac{9}{9}$ .

$- \frac{125}{27} + \frac{25 \cdot 3}{9 \cdot 3} + \frac{25 \cdot 9}{3 \cdot 9} + 8$

Step 1.4.2.2.2.5

Write $8$ as a fraction with denominator $1$ .

$- \frac{125}{27} + \frac{25 \cdot 3}{9 \cdot 3} + \frac{25 \cdot 9}{3 \cdot 9} + \frac{8}{1}$

Step 1.4.2.2.2.6

Multiply $\frac{8}{1}$ by $\frac{27}{27}$ .

$- \frac{125}{27} + \frac{25 \cdot 3}{9 \cdot 3} + \frac{25 \cdot 9}{3 \cdot 9} + \frac{8}{1} \cdot \frac{27}{27}$

Step 1.4.2.2.2.7

Multiply $\frac{8}{1}$ by $\frac{27}{27}$ .

$- \frac{125}{27} + \frac{25 \cdot 3}{9 \cdot 3} + \frac{25 \cdot 9}{3 \cdot 9} + \frac{8 \cdot 27}{27}$

Step 1.4.2.2.2.8

Reorder the factors of $9 \cdot 3$ .

$- \frac{125}{27} + \frac{25 \cdot 3}{3 \cdot 9} + \frac{25 \cdot 9}{3 \cdot 9} + \frac{8 \cdot 27}{27}$

Step 1.4.2.2.2.9

Multiply $3$ by $9$ .

$- \frac{125}{27} + \frac{25 \cdot 3}{27} + \frac{25 \cdot 9}{3 \cdot 9} + \frac{8 \cdot 27}{27}$

Step 1.4.2.2.2.10

Multiply $3$ by $9$ .

$- \frac{125}{27} + \frac{25 \cdot 3}{27} + \frac{25 \cdot 9}{27} + \frac{8 \cdot 27}{27}$

Step 1.4.2.2.3

Combine the numerators over the common denominator.

$\frac{- 125 + 25 \cdot 3 + 25 \cdot 9 + 8 \cdot 27}{27}$

Step 1.4.2.2.4

Simplify each term.

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Step 1.4.2.2.4.1

Multiply $25$ by $3$ .

$\frac{- 125 + 75 + 25 \cdot 9 + 8 \cdot 27}{27}$

Step 1.4.2.2.4.2

Multiply $25$ by $9$ .

$\frac{- 125 + 75 + 225 + 8 \cdot 27}{27}$

Step 1.4.2.2.4.3

Multiply $8$ by $27$ .

$\frac{- 125 + 75 + 225 + 216}{27}$

Step 1.4.2.2.5

Simplify by adding numbers.

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Step 1.4.2.2.5.1

Add $- 125$ and $75$ .

$\frac{- 50 + 225 + 216}{27}$

Step 1.4.2.2.5.2

Add $- 50$ and $225$ .

$\frac{175 + 216}{27}$

Step 1.4.2.2.5.3

Add $175$ and $216$ .

$\frac{391}{27}$

Step 1.4.3

List all of the points.

$(1, 5), (- \frac{5}{3}, \frac{391}{27})$

Step 2

Exclude the points that are not on the interval.

$(1, 5)$

Step 3

Use the first derivative test to determine which points can be maxima or minima.

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Step 3.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, - \frac{5}{3}) \cup (- \frac{5}{3}, 1) \cup (1, \infty)$

Step 3.2

Substitute any number, such as $- 4$ , from the interval $(- \infty, - \frac{5}{3})$ in the first derivative $3 x^{2} + 2 x - 5$ to check if the result is negative or positive.

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Step 3.2.1

Replace the variable $x$ with $- 4$ in the expression.

$f' (- 4) = 3 {(- 4)}^{2} + 2 (- 4) - 5$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

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

$f' (- 4) = 3 \cdot 16 + 2 (- 4) - 5$

Step 3.2.2.1.2

Multiply $3$ by $16$ .

$f' (- 4) = 48 + 2 (- 4) - 5$

Step 3.2.2.1.3

Multiply $2$ by $- 4$ .

$f' (- 4) = 48 - 8 - 5$

Step 3.2.2.2

Simplify by subtracting numbers.

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Step 3.2.2.2.1

Subtract $8$ from $48$ .

$f' (- 4) = 40 - 5$

Step 3.2.2.2.2

Subtract $5$ from $40$ .

$f' (- 4) = 35$

Step 3.2.2.3

The final answer is $35$ .

$35$

Step 3.3

Substitute any number, such as $0$ , from the interval $(- \frac{5}{3}, 1)$ in the first derivative $3 x^{2} + 2 x - 5$ to check if the result is negative or positive.

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Step 3.3.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = 3 {(0)}^{2} + 2 (0) - 5$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Raising $0$ to any positive power yields $0$ .

$f' (0) = 3 \cdot 0 + 2 (0) - 5$

Step 3.3.2.1.2

Multiply $3$ by $0$ .

$f' (0) = 0 + 2 (0) - 5$

Step 3.3.2.1.3

Multiply $2$ by $0$ .

$f' (0) = 0 + 0 - 5$

Step 3.3.2.2

Simplify by adding and subtracting.

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Step 3.3.2.2.1

Add $0$ and $0$ .

$f' (0) = 0 - 5$

Step 3.3.2.2.2

Subtract $5$ from $0$ .

$f' (0) = - 5$

Step 3.3.2.3

The final answer is $- 5$ .

$- 5$

Step 3.4

Substitute any number, such as $4$ , from the interval $(1, \infty)$ in the first derivative $3 x^{2} + 2 x - 5$ to check if the result is negative or positive.

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Step 3.4.1

Replace the variable $x$ with $4$ in the expression.

$f' (4) = 3 {(4)}^{2} + 2 (4) - 5$

Step 3.4.2

Simplify the result.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

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

$f' (4) = 3 \cdot 16 + 2 (4) - 5$

Step 3.4.2.1.2

Multiply $3$ by $16$ .

$f' (4) = 48 + 2 (4) - 5$

Step 3.4.2.1.3

Multiply $2$ by $4$ .

$f' (4) = 48 + 8 - 5$

Step 3.4.2.2

Simplify by adding and subtracting.

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Step 3.4.2.2.1

Add $48$ and $8$ .

$f' (4) = 56 - 5$

Step 3.4.2.2.2

Subtract $5$ from $56$ .

$f' (4) = 51$

Step 3.4.2.3

The final answer is $51$ .

$51$

Step 3.5

Since the first derivative changed signs from positive to negative around $x = - \frac{5}{3}$ , then $x = - \frac{5}{3}$ is a local maximum.

$x = - \frac{5}{3}$ is a local maximum

Step 3.6

Since the first derivative changed signs from negative to positive around $x = 1$ , then $x = 1$ is a local minimum.

$x = 1$ is a local minimum

Step 3.7

These are the local extrema for $f (x) = x^{3} + x^{2} - 5 x + 8$ .

$x = - \frac{5}{3}$ is a local maximum

$x = 1$ is a local minimum

$x = - \frac{5}{3}$ is a local maximum

$x = 1$ is a local minimum

Step 4

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

No absolute maximum

Absolute Minimum: $(1, 5)$

Step 5