Calculus Examples

$f (x) = 4 x^{3} - 45 x^{2} + 150 x$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $4 x^{3} - 45 x^{2} + 150 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 45 x^{2}] + \frac{d}{d x} [150 x]$ .

$f' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 45 x^{2}) + \frac{d}{d x} (150 x)$

Step 1.1.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 1.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 45 x^{2}) + \frac{d}{d x} (150 x)$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = 4 (3 x^{2}) + \frac{d}{d x} (- 45 x^{2}) + \frac{d}{d x} (150 x)$

Step 1.1.2.3

Multiply $3$ by $4$ .

$f' (x) = 12 x^{2} + \frac{d}{d x} (- 45 x^{2}) + \frac{d}{d x} (150 x)$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 45 x^{2}]$ .

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Step 1.1.3.1

Since $- 45$ is constant with respect to $x$ , the derivative of $- 45 x^{2}$ with respect to $x$ is $- 45 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 12 x^{2} - 45 \frac{d}{d x} x^{2} + \frac{d}{d x} (150 x)$

Step 1.1.3.2

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

$f' (x) = 12 x^{2} - 45 (2 x) + \frac{d}{d x} (150 x)$

Step 1.1.3.3

Multiply $2$ by $- 45$ .

$f' (x) = 12 x^{2} - 90 x + \frac{d}{d x} (150 x)$

Step 1.1.4

Evaluate $\frac{d}{d x} [150 x]$ .

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Step 1.1.4.1

Since $150$ is constant with respect to $x$ , the derivative of $150 x$ with respect to $x$ is $150 \frac{d}{d x} [x]$ .

$f' (x) = 12 x^{2} - 90 x + 150 \frac{d}{d x} (x)$

Step 1.1.4.2

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

$f' (x) = 12 x^{2} - 90 x + 150 \cdot 1$

Step 1.1.4.3

Multiply $150$ by $1$ .

$f' (x) = 12 x^{2} - 90 x + 150$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 90 x + 150$ .

$12 x^{2} - 90 x + 150$

Step 2

Set the first derivative equal to $0$ then solve the equation $12 x^{2} - 90 x + 150 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$12 x^{2} - 90 x + 150 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $6$ out of $12 x^{2} - 90 x + 150$ .

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Step 2.2.1.1

Factor $6$ out of $12 x^{2}$ .

$6 (2 x^{2}) - 90 x + 150 = 0$

Step 2.2.1.2

Factor $6$ out of $- 90 x$ .

$6 (2 x^{2}) + 6 (- 15 x) + 150 = 0$

Step 2.2.1.3

Factor $6$ out of $150$ .

$6 (2 x^{2}) + 6 (- 15 x) + 6 (25) = 0$

Step 2.2.1.4

Factor $6$ out of $6 (2 x^{2}) + 6 (- 15 x)$ .

$6 (2 x^{2} - 15 x) + 6 (25) = 0$

Step 2.2.1.5

Factor $6$ out of $6 (2 x^{2} - 15 x) + 6 (25)$ .

$6 (2 x^{2} - 15 x + 25) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor by grouping.

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Step 2.2.2.1.1

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 $a \cdot c = 2 \cdot 25 = 50$ and whose sum is $b = - 15$ .

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Step 2.2.2.1.1.1

Factor $- 15$ out of $- 15 x$ .

$6 (2 x^{2} - 15 x + 25) = 0$

Step 2.2.2.1.1.2

Rewrite $- 15$ as $- 5$ plus $- 10$

$6 (2 x^{2} + (- 5 - 10) x + 25) = 0$

Step 2.2.2.1.1.3

Apply the distributive property.

$6 (2 x^{2} - 5 x - 10 x + 25) = 0$

Step 2.2.2.1.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1.2.1

Group the first two terms and the last two terms.

$6 ((2 x^{2} - 5 x) - 10 x + 25) = 0$

Step 2.2.2.1.2.2

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

$6 (x (2 x - 5) - 5 (2 x - 5)) = 0$

Step 2.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 5$ .

$6 ((2 x - 5) (x - 5)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$6 (2 x - 5) (x - 5) = 0$

Step 2.3

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

$2 x - 5 = 0$

$x - 5 = 0$

Step 2.4

Set $2 x - 5$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $2 x - 5$ equal to $0$ .

$2 x - 5 = 0$

Step 2.4.2

Solve $2 x - 5 = 0$ for $x$ .

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Step 2.4.2.1

Add $5$ to both sides of the equation.

$2 x = 5$

Step 2.4.2.2

Divide each term in $2 x = 5$ by $2$ and simplify.

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Step 2.4.2.2.1

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{2}$

Step 2.5

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.5.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.6

The final solution is all the values that make $6 (2 x - 5) (x - 5) = 0$ true.

$x = \frac{5}{2}, 5$

Step 3

The values which make the derivative equal to $0$ are $\frac{5}{2}, 5$ .

$\frac{5}{2}, 5$

Step 4

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

$(- \infty, \frac{5}{2}) \cup (\frac{5}{2}, 5) \cup (5, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{5}{2})$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $1.5$ in the expression.

$f' (1.5) = 12 {(1.5)}^{2} - 90 \cdot 1.5 + 150$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $1.5$ to the power of $2$ .

$f' (1.5) = 12 \cdot 2.25 - 90 \cdot 1.5 + 150$

Step 5.2.1.2

Multiply $12$ by $2.25$ .

$f' (1.5) = 27 - 90 \cdot 1.5 + 150$

Step 5.2.1.3

Multiply $- 90$ by $1.5$ .

$f' (1.5) = 27 - 135 + 150$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $135$ from $27$ .

$f' (1.5) = - 108 + 150$

Step 5.2.2.2

Add $- 108$ and $150$ .

$f' (1.5) = 42$

Step 5.2.3

The final answer is $42$ .

$42$

Step 5.3

At $x = 1.5$ the derivative is $42$ . Since this is positive, the function is increasing on $(- \infty, \frac{5}{2})$ .

Increasing on $(- \infty, \frac{5}{2})$ since $f' (x) > 0$

Step 6

Substitute a value from the interval $(2.5, 5)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $3.75$ in the expression.

$f' (3.75) = 12 {(3.75)}^{2} - 90 \cdot 3.75 + 150$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $3.75$ to the power of $2$ .

$f' (3.75) = 12 \cdot 14.0625 - 90 \cdot 3.75 + 150$

Step 6.2.1.2

Multiply $12$ by $14.0625$ .

$f' (3.75) = 168.75 - 90 \cdot 3.75 + 150$

Step 6.2.1.3

Multiply $- 90$ by $3.75$ .

$f' (3.75) = 168.75 - 337.5 + 150$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $337.5$ from $168.75$ .

$f' (3.75) = - 168.75 + 150$

Step 6.2.2.2

Add $- 168.75$ and $150$ .

$f' (3.75) = - 18.75$

Step 6.2.3

The final answer is $- 18.75$ .

$- 18.75$

Step 6.3

At $x = 3.75$ the derivative is $- 18.75$ . Since this is negative, the function is decreasing on $(2.5, 5)$ .

Decreasing on $(\frac{5}{2}, 5)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(5, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $6$ in the expression.

$f' (6) = 12 {(6)}^{2} - 90 \cdot 6 + 150$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $6$ to the power of $2$ .

$f' (6) = 12 \cdot 36 - 90 \cdot 6 + 150$

Step 7.2.1.2

Multiply $12$ by $36$ .

$f' (6) = 432 - 90 \cdot 6 + 150$

Step 7.2.1.3

Multiply $- 90$ by $6$ .

$f' (6) = 432 - 540 + 150$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $540$ from $432$ .

$f' (6) = - 108 + 150$

Step 7.2.2.2

Add $- 108$ and $150$ .

$f' (6) = 42$

Step 7.2.3

The final answer is $42$ .

$42$

Step 7.3

At $x = 6$ the derivative is $42$ . Since this is positive, the function is increasing on $(5, \infty)$ .

Increasing on $(5, \infty)$ since $f' (x) > 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, \frac{5}{2}), (5, \infty)$

Decreasing on: $(\frac{5}{2}, 5)$

Step 9