Calculus Examples

$f (x) = 4 x^{3} + 3 x^{2} - 6 x$

Step 1

Find the second derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $4 x^{3} + 3 x^{2} - 6 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 6 x]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 6 x]$

Step 1.1.2

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

Tap for more steps...

Step 1.1.2.1

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

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 6 x]$

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 $n = 3$ .

$4 (3 x^{2}) + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 6 x]$

Step 1.1.2.3

Multiply $3$ by $4$ .

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

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

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

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

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 $n = 2$ .

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

Step 1.1.3.3

Multiply $2$ by $3$ .

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

Step 1.1.4

Evaluate $\frac{d}{d x} [- 6 x]$ .

Tap for more steps...

Step 1.1.4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

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

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 $n = 1$ .

$12 x^{2} + 6 x - 6 \cdot 1$

Step 1.1.4.3

Multiply $- 6$ by $1$ .

$f' (x) = 12 x^{2} + 6 x - 6$

Step 1.2

Find the second derivative.

Tap for more steps...

Step 1.2.1

By the Sum Rule, the derivative of $12 x^{2} + 6 x - 6$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 6]$ .

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

Step 1.2.2

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

Tap for more steps...

Step 1.2.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

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

Step 1.2.2.2

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

$12 (2 x) + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 6]$

Step 1.2.2.3

Multiply $2$ by $12$ .

$24 x + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 6]$

Step 1.2.3

Evaluate $\frac{d}{d x} [6 x]$ .

Tap for more steps...

Step 1.2.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$24 x + 6 \frac{d}{d x} [x] + \frac{d}{d x} [- 6]$

Step 1.2.3.2

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

$24 x + 6 \cdot 1 + \frac{d}{d x} [- 6]$

Step 1.2.3.3

Multiply $6$ by $1$ .

$24 x + 6 + \frac{d}{d x} [- 6]$

Step 1.2.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.2.4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$24 x + 6 + 0$

Step 1.2.4.2

Add $24 x + 6$ and $0$ .

$f'' (x) = 24 x + 6$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $24 x + 6$ .

$24 x + 6$

Step 2

Set the second derivative equal to $0$ then solve the equation $24 x + 6 = 0$ .

Tap for more steps...

Step 2.1

Set the second derivative equal to $0$ .

$24 x + 6 = 0$

Step 2.2

Subtract $6$ from both sides of the equation.

$24 x = - 6$

Step 2.3

Divide each term in $24 x = - 6$ by $24$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $24 x = - 6$ by $24$ .

$\frac{24 x}{24} = \frac{- 6}{24}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $24$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$\frac{24 x}{24} = \frac{- 6}{24}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{24}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Cancel the common factor of $- 6$ and $24$ .

Tap for more steps...

Step 2.3.3.1.1

Factor $6$ out of $- 6$ .

$x = \frac{6 (- 1)}{24}$

Step 2.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.3.1.2.1

Factor $6$ out of $24$ .

$x = \frac{6 \cdot - 1}{6 \cdot 4}$

Step 2.3.3.1.2.2

Cancel the common factor.

$x = \frac{6 \cdot - 1}{6 \cdot 4}$

Step 2.3.3.1.2.3

Rewrite the expression.

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

Step 2.3.3.2

Move the negative in front of the fraction.

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

Step 3

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

Tap for more steps...

Step 3.1

Substitute $- \frac{1}{4}$ in $f (x) = 4 x^{3} + 3 x^{2} - 6 x$ to find the value of $y$ .

Tap for more steps...

Step 3.1.1

Replace the variable $x$ with $- \frac{1}{4}$ in the expression.

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

Step 3.1.2

Simplify the result.

Tap for more steps...

Step 3.1.2.1

Simplify each term.

Tap for more steps...

Step 3.1.2.1.1

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

Tap for more steps...

Step 3.1.2.1.1.1

Apply the product rule to $- \frac{1}{4}$ .

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

Step 3.1.2.1.1.2

Apply the product rule to $\frac{1}{4}$ .

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

Step 3.1.2.1.2

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

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

Step 3.1.2.1.3

One to any power is one.

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

Step 3.1.2.1.4

Raise $4$ to the power of $3$ .

$f (- \frac{1}{4}) = 4 (- \frac{1}{64}) + 3 {(- \frac{1}{4})}^{2} - 6 (- \frac{1}{4})$

Step 3.1.2.1.5

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.1.2.1.5.1

Move the leading negative in $- \frac{1}{64}$ into the numerator.

$f (- \frac{1}{4}) = 4 (\frac{- 1}{64}) + 3 {(- \frac{1}{4})}^{2} - 6 (- \frac{1}{4})$

Step 3.1.2.1.5.2

Factor $4$ out of $64$ .

$f (- \frac{1}{4}) = 4 (\frac{- 1}{4 (16)}) + 3 {(- \frac{1}{4})}^{2} - 6 (- \frac{1}{4})$

Step 3.1.2.1.5.3

Cancel the common factor.

$f (- \frac{1}{4}) = 4 (\frac{- 1}{4 \cdot 16}) + 3 {(- \frac{1}{4})}^{2} - 6 (- \frac{1}{4})$

Step 3.1.2.1.5.4

Rewrite the expression.

$f (- \frac{1}{4}) = \frac{- 1}{16} + 3 {(- \frac{1}{4})}^{2} - 6 (- \frac{1}{4})$

Step 3.1.2.1.6

Move the negative in front of the fraction.

$f (- \frac{1}{4}) = - \frac{1}{16} + 3 {(- \frac{1}{4})}^{2} - 6 (- \frac{1}{4})$

Step 3.1.2.1.7

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

Tap for more steps...

Step 3.1.2.1.7.1

Apply the product rule to $- \frac{1}{4}$ .

$f (- \frac{1}{4}) = - \frac{1}{16} + 3 ({(- 1)}^{2} {(\frac{1}{4})}^{2}) - 6 (- \frac{1}{4})$

Step 3.1.2.1.7.2

Apply the product rule to $\frac{1}{4}$ .

$f (- \frac{1}{4}) = - \frac{1}{16} + 3 ({(- 1)}^{2} (\frac{1^{2}}{4^{2}})) - 6 (- \frac{1}{4})$

Step 3.1.2.1.8

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

$f (- \frac{1}{4}) = - \frac{1}{16} + 3 (1 (\frac{1^{2}}{4^{2}})) - 6 (- \frac{1}{4})$

Step 3.1.2.1.9

Multiply $\frac{1^{2}}{4^{2}}$ by $1$ .

$f (- \frac{1}{4}) = - \frac{1}{16} + 3 (\frac{1^{2}}{4^{2}}) - 6 (- \frac{1}{4})$

Step 3.1.2.1.10

One to any power is one.

$f (- \frac{1}{4}) = - \frac{1}{16} + 3 (\frac{1}{4^{2}}) - 6 (- \frac{1}{4})$

Step 3.1.2.1.11

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

$f (- \frac{1}{4}) = - \frac{1}{16} + 3 (\frac{1}{16}) - 6 (- \frac{1}{4})$

Step 3.1.2.1.12

Combine $3$ and $\frac{1}{16}$ .

$f (- \frac{1}{4}) = - \frac{1}{16} + \frac{3}{16} - 6 (- \frac{1}{4})$

Step 3.1.2.1.13

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.1.2.1.13.1

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

$f (- \frac{1}{4}) = - \frac{1}{16} + \frac{3}{16} - 6 (\frac{- 1}{4})$

Step 3.1.2.1.13.2

Factor $2$ out of $- 6$ .

$f (- \frac{1}{4}) = - \frac{1}{16} + \frac{3}{16} + 2 (- 3) (\frac{- 1}{4})$

Step 3.1.2.1.13.3

Factor $2$ out of $4$ .

$f (- \frac{1}{4}) = - \frac{1}{16} + \frac{3}{16} + 2 \cdot (- 3 \frac{- 1}{2 \cdot 2})$

Step 3.1.2.1.13.4

Cancel the common factor.

$f (- \frac{1}{4}) = - \frac{1}{16} + \frac{3}{16} + 2 \cdot (- 3 \frac{- 1}{2 \cdot 2})$

Step 3.1.2.1.13.5

Rewrite the expression.

$f (- \frac{1}{4}) = - \frac{1}{16} + \frac{3}{16} - 3 (\frac{- 1}{2})$

Step 3.1.2.1.14

Combine $- 3$ and $\frac{- 1}{2}$ .

$f (- \frac{1}{4}) = - \frac{1}{16} + \frac{3}{16} + \frac{- 3 \cdot - 1}{2}$

Step 3.1.2.1.15

Multiply $- 3$ by $- 1$ .

$f (- \frac{1}{4}) = - \frac{1}{16} + \frac{3}{16} + \frac{3}{2}$

Step 3.1.2.2

Simplify terms.

Tap for more steps...

Step 3.1.2.2.1

Combine the numerators over the common denominator.

$f (- \frac{1}{4}) = \frac{- 1 + 3}{16} + \frac{3}{2}$

Step 3.1.2.2.2

Add $- 1$ and $3$ .

$f (- \frac{1}{4}) = \frac{2}{16} + \frac{3}{2}$

Step 3.1.2.2.3

Cancel the common factor of $2$ and $16$ .

Tap for more steps...

Step 3.1.2.2.3.1

Factor $2$ out of $2$ .

$f (- \frac{1}{4}) = \frac{2 (1)}{16} + \frac{3}{2}$

Step 3.1.2.2.3.2

Cancel the common factors.

Tap for more steps...

Step 3.1.2.2.3.2.1

Factor $2$ out of $16$ .

$f (- \frac{1}{4}) = \frac{2 \cdot 1}{2 \cdot 8} + \frac{3}{2}$

Step 3.1.2.2.3.2.2

Cancel the common factor.

$f (- \frac{1}{4}) = \frac{2 \cdot 1}{2 \cdot 8} + \frac{3}{2}$

Step 3.1.2.2.3.2.3

Rewrite the expression.

$f (- \frac{1}{4}) = \frac{1}{8} + \frac{3}{2}$

Step 3.1.2.3

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (- \frac{1}{4}) = \frac{1}{8} + \frac{3}{2} \cdot \frac{4}{4}$

Step 3.1.2.4

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1.2.4.1

Multiply $\frac{3}{2}$ by $\frac{4}{4}$ .

$f (- \frac{1}{4}) = \frac{1}{8} + \frac{3 \cdot 4}{2 \cdot 4}$

Step 3.1.2.4.2

Multiply $2$ by $4$ .

$f (- \frac{1}{4}) = \frac{1}{8} + \frac{3 \cdot 4}{8}$

Step 3.1.2.5

Combine the numerators over the common denominator.

$f (- \frac{1}{4}) = \frac{1 + 3 \cdot 4}{8}$

Step 3.1.2.6

Simplify the numerator.

Tap for more steps...

Step 3.1.2.6.1

Multiply $3$ by $4$ .

$f (- \frac{1}{4}) = \frac{1 + 12}{8}$

Step 3.1.2.6.2

Add $1$ and $12$ .

$f (- \frac{1}{4}) = \frac{13}{8}$

Step 3.1.2.7

The final answer is $\frac{13}{8}$ .

$\frac{13}{8}$

Step 3.2

The point found by substituting $- \frac{1}{4}$ in $f (x) = 4 x^{3} + 3 x^{2} - 6 x$ is $(- \frac{1}{4}, \frac{13}{8})$ . This point can be an inflection point.

$(- \frac{1}{4}, \frac{13}{8})$

Step 4

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

$(- \infty, - \frac{1}{4}) \cup (- \frac{1}{4}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \frac{1}{4})$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $- 0.35$ in the expression.

$f'' (- 0.35) = 24 (- 0.35) + 6$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Multiply $24$ by $- 0.35$ .

$f'' (- 0.35) = - 8.4 + 6$

Step 5.2.2

Add $- 8.4$ and $6$ .

$f'' (- 0.35) = - 2.4$

Step 5.2.3

The final answer is $- 2.4$ .

$- 2.4$

Step 5.3

At $- 0.35$ , the second derivative is $- 2.4$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - \frac{1}{4})$

Decreasing on $(- \infty, - \frac{1}{4})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- \frac{1}{4}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $- 0.15$ in the expression.

$f'' (- 0.15) = 24 (- 0.15) + 6$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Multiply $24$ by $- 0.15$ .

$f'' (- 0.15) = - 3.6 + 6$

Step 6.2.2

Add $- 3.6$ and $6$ .

$f'' (- 0.15) = 2.4$

Step 6.2.3

The final answer is $2.4$ .

$2.4$

Step 6.3

At $- 0.15$ , the second derivative is $2.4$ . Since this is positive, the second derivative is increasing on the interval $(- \frac{1}{4}, \infty)$ .

Increasing on $(- \frac{1}{4}, \infty)$ since $f'' (x) > 0$

Step 7

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 $(- \frac{1}{4}, \frac{13}{8})$ .

$(- \frac{1}{4}, \frac{13}{8})$

Step 8