Calculus Examples

$\frac{2}{3} x^{3} - x^{2} + 12 x + 20$

Step 1

Write $\frac{2}{3} x^{3} - x^{2} + 12 x + 20$ as a function.

$f (x) = \frac{2}{3} x^{3} - x^{2} + 12 x + 20$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Find the first derivative.

Tap for more steps...

Step 2.1.1

By the Sum Rule, the derivative of $\frac{2}{3} x^{3} - x^{2} + 12 x + 20$ with respect to $x$ is $\frac{d}{d x} [\frac{2}{3} x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$ .

$\frac{d}{d x} [\frac{2}{3} x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.2

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

Tap for more steps...

Step 2.1.2.1

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

$\frac{2}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.2.2

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

$\frac{2}{3} (3 x^{2}) + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.2.3

Combine $3$ and $\frac{2}{3}$ .

$\frac{3 \cdot 2}{3} x^{2} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.2.4

Multiply $3$ by $2$ .

$\frac{6}{3} x^{2} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.2.5

Combine $\frac{6}{3}$ and $x^{2}$ .

$\frac{6 x^{2}}{3} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.2.6

Cancel the common factor of $6$ and $3$ .

Tap for more steps...

Step 2.1.2.6.1

Factor $3$ out of $6 x^{2}$ .

$\frac{3 (2 x^{2})}{3} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.2.6.2

Cancel the common factors.

Tap for more steps...

Step 2.1.2.6.2.1

Factor $3$ out of $3$ .

$\frac{3 (2 x^{2})}{3 (1)} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.2.6.2.2

Cancel the common factor.

$\frac{3 (2 x^{2})}{3 \cdot 1} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.2.6.2.3

Rewrite the expression.

$\frac{2 x^{2}}{1} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.2.6.2.4

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

$2 x^{2} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.3

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

Tap for more steps...

Step 2.1.3.1

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

$2 x^{2} - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.3.2

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

$2 x^{2} - (2 x) + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.3.3

Multiply $2$ by $- 1$ .

$2 x^{2} - 2 x + \frac{d}{d x} [12 x] + \frac{d}{d x} [20]$

Step 2.1.4

Evaluate $\frac{d}{d x} [12 x]$ .

Tap for more steps...

Step 2.1.4.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$2 x^{2} - 2 x + 12 \frac{d}{d x} [x] + \frac{d}{d x} [20]$

Step 2.1.4.2

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

$2 x^{2} - 2 x + 12 \cdot 1 + \frac{d}{d x} [20]$

Step 2.1.4.3

Multiply $12$ by $1$ .

$2 x^{2} - 2 x + 12 + \frac{d}{d x} [20]$

Step 2.1.5

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.1.5.1

Since $20$ is constant with respect to $x$ , the derivative of $20$ with respect to $x$ is $0$ .

$2 x^{2} - 2 x + 12 + 0$

Step 2.1.5.2

Add $2 x^{2} - 2 x + 12$ and $0$ .

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

Step 2.2

Find the second derivative.

Tap for more steps...

Step 2.2.1

By the Sum Rule, the derivative of $2 x^{2} - 2 x + 12$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [12]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [12]$

Step 2.2.2

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

Tap for more steps...

Step 2.2.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [12]$

Step 2.2.2.2

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

$2 (2 x) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [12]$

Step 2.2.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [12]$

Step 2.2.3

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

Tap for more steps...

Step 2.2.3.1

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

$4 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [12]$

Step 2.2.3.2

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

$4 x - 2 \cdot 1 + \frac{d}{d x} [12]$

Step 2.2.3.3

Multiply $- 2$ by $1$ .

$4 x - 2 + \frac{d}{d x} [12]$

Step 2.2.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.2.4.1

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$4 x - 2 + 0$

Step 2.2.4.2

Add $4 x - 2$ and $0$ .

$f'' (x) = 4 x - 2$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $4 x - 2$ .

$4 x - 2$

Step 3

Set the second derivative equal to $0$ then solve the equation $4 x - 2 = 0$ .

Tap for more steps...

Step 3.1

Set the second derivative equal to $0$ .

$4 x - 2 = 0$

Step 3.2

Add $2$ to both sides of the equation.

$4 x = 2$

Step 3.3

Divide each term in $4 x = 2$ by $4$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $4 x = 2$ by $4$ .

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

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $x$ by $1$ .

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

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

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

Tap for more steps...

Step 3.3.3.1.1

Factor $2$ out of $2$ .

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

Step 3.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.3.3.1.2.1

Factor $2$ out of $4$ .

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

Step 3.3.3.1.2.2

Cancel the common factor.

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

Step 3.3.3.1.2.3

Rewrite the expression.

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

Step 4

Find the points where the second derivative is $0$ .

Tap for more steps...

Step 4.1

Substitute $\frac{1}{2}$ in $f (x) = \frac{2}{3} x^{3} - x^{2} + 12 x + 20$ to find the value of $y$ .

Tap for more steps...

Step 4.1.1

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

$f (\frac{1}{2}) = \frac{2}{3} \cdot {(\frac{1}{2})}^{3} - {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 20$

Step 4.1.2

Simplify the result.

Tap for more steps...

Step 4.1.2.1

Simplify each term.

Tap for more steps...

Step 4.1.2.1.1

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

$f (\frac{1}{2}) = \frac{2}{3} \cdot \frac{1^{3}}{2^{3}} - {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 20$

Step 4.1.2.1.2

Combine.

$f (\frac{1}{2}) = \frac{2 \cdot 1^{3}}{3 \cdot 2^{3}} - {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 20$

Step 4.1.2.1.3

Cancel the common factor of $2$ and $2^{3}$ .

Tap for more steps...

Step 4.1.2.1.3.1

Factor $2$ out of $2 \cdot 1^{3}$ .

$f (\frac{1}{2}) = \frac{2 (1^{3})}{3 \cdot 2^{3}} - {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 20$

Step 4.1.2.1.3.2

Cancel the common factors.

Tap for more steps...

Step 4.1.2.1.3.2.1

Factor $2$ out of $3 \cdot 2^{3}$ .

$f (\frac{1}{2}) = \frac{2 (1^{3})}{2 (3 \cdot 2^{2})} - {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 20$

Step 4.1.2.1.3.2.2

Cancel the common factor.

$f (\frac{1}{2}) = \frac{2 \cdot 1^{3}}{2 (3 \cdot 2^{2})} - {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 20$

Step 4.1.2.1.3.2.3

Rewrite the expression.

$f (\frac{1}{2}) = \frac{1^{3}}{3 \cdot 2^{2}} - {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 20$

Step 4.1.2.1.4

One to any power is one.

$f (\frac{1}{2}) = \frac{1}{3 \cdot 2^{2}} - {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 20$

Step 4.1.2.1.5

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

$f (\frac{1}{2}) = \frac{1}{3 \cdot 4} - {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 20$

Step 4.1.2.1.6

Multiply $3$ by $4$ .

$f (\frac{1}{2}) = \frac{1}{12} - {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 20$

Step 4.1.2.1.7

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

$f (\frac{1}{2}) = \frac{1}{12} - \frac{1^{2}}{2^{2}} + 12 (\frac{1}{2}) + 20$

Step 4.1.2.1.8

One to any power is one.

$f (\frac{1}{2}) = \frac{1}{12} - \frac{1}{2^{2}} + 12 (\frac{1}{2}) + 20$

Step 4.1.2.1.9

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

$f (\frac{1}{2}) = \frac{1}{12} - \frac{1}{4} + 12 (\frac{1}{2}) + 20$

Step 4.1.2.1.10

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.2.1.10.1

Factor $2$ out of $12$ .

$f (\frac{1}{2}) = \frac{1}{12} - \frac{1}{4} + 2 (6) (\frac{1}{2}) + 20$

Step 4.1.2.1.10.2

Cancel the common factor.

$f (\frac{1}{2}) = \frac{1}{12} - \frac{1}{4} + 2 \cdot (6 (\frac{1}{2})) + 20$

Step 4.1.2.1.10.3

Rewrite the expression.

$f (\frac{1}{2}) = \frac{1}{12} - \frac{1}{4} + 6 + 20$

Step 4.1.2.2

Find the common denominator.

Tap for more steps...

Step 4.1.2.2.1

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$f (\frac{1}{2}) = \frac{1}{12} - (\frac{1}{4} \cdot \frac{3}{3}) + 6 + 20$

Step 4.1.2.2.2

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$f (\frac{1}{2}) = \frac{1}{12} - \frac{3}{4 \cdot 3} + 6 + 20$

Step 4.1.2.2.3

Write $6$ as a fraction with denominator $1$ .

$f (\frac{1}{2}) = \frac{1}{12} - \frac{3}{4 \cdot 3} + \frac{6}{1} + 20$

Step 4.1.2.2.4

Multiply $\frac{6}{1}$ by $\frac{12}{12}$ .

$f (\frac{1}{2}) = \frac{1}{12} - \frac{3}{4 \cdot 3} + \frac{6}{1} \cdot \frac{12}{12} + 20$

Step 4.1.2.2.5

Multiply $\frac{6}{1}$ by $\frac{12}{12}$ .

$f (\frac{1}{2}) = \frac{1}{12} - \frac{3}{4 \cdot 3} + \frac{6 \cdot 12}{12} + 20$

Step 4.1.2.2.6

Write $20$ as a fraction with denominator $1$ .

$f (\frac{1}{2}) = \frac{1}{12} - \frac{3}{4 \cdot 3} + \frac{6 \cdot 12}{12} + \frac{20}{1}$

Step 4.1.2.2.7

Multiply $\frac{20}{1}$ by $\frac{12}{12}$ .

$f (\frac{1}{2}) = \frac{1}{12} - \frac{3}{4 \cdot 3} + \frac{6 \cdot 12}{12} + \frac{20}{1} \cdot \frac{12}{12}$

Step 4.1.2.2.8

Multiply $\frac{20}{1}$ by $\frac{12}{12}$ .

$f (\frac{1}{2}) = \frac{1}{12} - \frac{3}{4 \cdot 3} + \frac{6 \cdot 12}{12} + \frac{20 \cdot 12}{12}$

Step 4.1.2.2.9

Reorder the factors of $4 \cdot 3$ .

$f (\frac{1}{2}) = \frac{1}{12} - \frac{3}{3 \cdot 4} + \frac{6 \cdot 12}{12} + \frac{20 \cdot 12}{12}$

Step 4.1.2.2.10

Multiply $3$ by $4$ .

$f (\frac{1}{2}) = \frac{1}{12} - \frac{3}{12} + \frac{6 \cdot 12}{12} + \frac{20 \cdot 12}{12}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$f (\frac{1}{2}) = \frac{1 - 3 + 6 \cdot 12 + 20 \cdot 12}{12}$

Step 4.1.2.4

Simplify each term.

Tap for more steps...

Step 4.1.2.4.1

Multiply $6$ by $12$ .

$f (\frac{1}{2}) = \frac{1 - 3 + 72 + 20 \cdot 12}{12}$

Step 4.1.2.4.2

Multiply $20$ by $12$ .

$f (\frac{1}{2}) = \frac{1 - 3 + 72 + 240}{12}$

Step 4.1.2.5

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.1.2.5.1

Subtract $3$ from $1$ .

$f (\frac{1}{2}) = \frac{- 2 + 72 + 240}{12}$

Step 4.1.2.5.2

Simplify by adding numbers.

Tap for more steps...

Step 4.1.2.5.2.1

Add $- 2$ and $72$ .

$f (\frac{1}{2}) = \frac{70 + 240}{12}$

Step 4.1.2.5.2.2

Add $70$ and $240$ .

$f (\frac{1}{2}) = \frac{310}{12}$

Step 4.1.2.5.3

Cancel the common factor of $310$ and $12$ .

Tap for more steps...

Step 4.1.2.5.3.1

Factor $2$ out of $310$ .

$f (\frac{1}{2}) = \frac{2 (155)}{12}$

Step 4.1.2.5.3.2

Cancel the common factors.

Tap for more steps...

Step 4.1.2.5.3.2.1

Factor $2$ out of $12$ .

$f (\frac{1}{2}) = \frac{2 \cdot 155}{2 \cdot 6}$

Step 4.1.2.5.3.2.2

Cancel the common factor.

$f (\frac{1}{2}) = \frac{2 \cdot 155}{2 \cdot 6}$

Step 4.1.2.5.3.2.3

Rewrite the expression.

$f (\frac{1}{2}) = \frac{155}{6}$

Step 4.1.2.6

The final answer is $\frac{155}{6}$ .

$\frac{155}{6}$

Step 4.2

The point found by substituting $\frac{1}{2}$ in $f (x) = \frac{2}{3} x^{3} - x^{2} + 12 x + 20$ is $(\frac{1}{2}, \frac{155}{6})$ . This point can be an inflection point.

$(\frac{1}{2}, \frac{155}{6})$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, \frac{1}{2})$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $0.4$ in the expression.

$f'' (0.4) = 4 (0.4) - 2$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Multiply $4$ by $0.4$ .

$f'' (0.4) = 1.6 - 2$

Step 6.2.2

Subtract $2$ from $1.6$ .

$f'' (0.4) = - 0.4$

Step 6.2.3

The final answer is $- 0.4$ .

$- 0.4$

Step 6.3

At $0.4$ , the second derivative is $- 0.4$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, \frac{1}{2})$

Decreasing on $(- \infty, \frac{1}{2})$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(\frac{1}{2}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $0.6$ in the expression.

$f'' (0.6) = 4 (0.6) - 2$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Multiply $4$ by $0.6$ .

$f'' (0.6) = 2.4 - 2$

Step 7.2.2

Subtract $2$ from $2.4$ .

$f'' (0.6) = 0.4$

Step 7.2.3

The final answer is $0.4$ .

$0.4$

Step 7.3

At $0.6$ , the second derivative is $0.4$ . Since this is positive, the second derivative is increasing on the interval $(\frac{1}{2}, \infty)$ .

Increasing on $(\frac{1}{2}, \infty)$ since $f'' (x) > 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(\frac{1}{2}, \frac{155}{6})$ .

$(\frac{1}{2}, \frac{155}{6})$

Step 9