Calculus Examples

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

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $3$ by $- 1$ .

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

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $2$ by $3$ .

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

Step 1.4

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

Tap for more steps...

Step 1.4.1

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

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

Step 1.4.2

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

$- 3 x^{2} + 6 x - 3 \cdot 1$

Step 1.4.3

Multiply $- 3$ by $1$ .

$- 3 x^{2} + 6 x - 3$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

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

$f'' (x) = \frac{d}{d x} (- 3 x^{2}) + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 3)$

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

$f'' (x) = - 3 \frac{d}{d x} x^{2} + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 3)$

Step 2.2.2

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

$f'' (x) = - 3 (2 x) + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 3)$

Step 2.2.3

Multiply $2$ by $- 3$ .

$f'' (x) = - 6 x + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 3)$

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

$f'' (x) = - 6 x + 6 \frac{d}{d x} (x) + \frac{d}{d x} (- 3)$

Step 2.3.2

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

$f'' (x) = - 6 x + 6 \cdot 1 + \frac{d}{d x} (- 3)$

Step 2.3.3

Multiply $6$ by $1$ .

$f'' (x) = - 6 x + 6 + \frac{d}{d x} (- 3)$

Step 2.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.4.1

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

$f'' (x) = - 6 x + 6 + 0$

Step 2.4.2

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

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

Step 3

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

$- 3 x^{2} + 6 x - 3 = 0$

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

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

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

Step 4.1.2

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

Tap for more steps...

Step 4.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

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

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 $n = 3$ .

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

Step 4.1.2.3

Multiply $3$ by $- 1$ .

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

Step 4.1.3

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

Tap for more steps...

Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

Multiply $2$ by $3$ .

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

Step 4.1.4

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

Tap for more steps...

Step 4.1.4.1

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

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

Step 4.1.4.2

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

$- 3 x^{2} + 6 x - 3 \cdot 1$

Step 4.1.4.3

Multiply $- 3$ by $1$ .

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- 3 x^{2} + 6 x - 3$ .

$- 3 x^{2} + 6 x - 3$

Step 5

Set the first derivative equal to $0$ then solve the equation $- 3 x^{2} + 6 x - 3 = 0$ .

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

$- 3 x^{2} + 6 x - 3 = 0$

Step 5.2

Factor the left side of the equation.

Tap for more steps...

Step 5.2.1

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

Tap for more steps...

Step 5.2.1.1

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

$- 3 x^{2} + 6 x - 3 = 0$

Step 5.2.1.2

Factor $- 3$ out of $6 x$ .

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

Step 5.2.1.3

Factor $- 3$ out of $- 3$ .

$- 3 x^{2} - 3 (- 2 x) - 3 \cdot 1 = 0$

Step 5.2.1.4

Factor $- 3$ out of $- 3 (x^{2}) - 3 (- 2 x)$ .

$- 3 (x^{2} - 2 x) - 3 \cdot 1 = 0$

Step 5.2.1.5

Factor $- 3$ out of $- 3 (x^{2} - 2 x) - 3 (1)$ .

$- 3 (x^{2} - 2 x + 1) = 0$

Step 5.2.2

Factor using the perfect square rule.

Tap for more steps...

Step 5.2.2.1

Rewrite $1$ as $1^{2}$ .

$- 3 (x^{2} - 2 x + 1^{2}) = 0$

Step 5.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 5.2.2.3

Rewrite the polynomial.

$- 3 (x^{2} - 2 \cdot x \cdot 1 + 1^{2}) = 0$

Step 5.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

$- 3 {(x - 1)}^{2} = 0$

Step 5.3

Divide each term in $- 3 {(x - 1)}^{2} = 0$ by $- 3$ and simplify.

Tap for more steps...

Step 5.3.1

Divide each term in $- 3 {(x - 1)}^{2} = 0$ by $- 3$ .

$\frac{- 3 {(x - 1)}^{2}}{- 3} = \frac{0}{- 3}$

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 3 {(x - 1)}^{2}}{- 3} = \frac{0}{- 3}$

Step 5.3.2.1.2

Divide ${(x - 1)}^{2}$ by $1$ .

${(x - 1)}^{2} = \frac{0}{- 3}$

Step 5.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.1

Divide $0$ by $- 3$ .

${(x - 1)}^{2} = 0$

Step 5.4

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

$x - 1 = 0$

Step 5.5

Add $1$ to both sides of the equation.

$x = 1$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

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 7

Critical points to evaluate.

$x = 1$

Step 8

Evaluate the second derivative at $x = 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 6 \cdot 1 + 6$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Multiply $- 6$ by $1$ .

$- 6 + 6$

Step 9.2

Add $- 6$ and $6$ .

$0$

Step 10

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

Tap for more steps...

Step 10.1

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

$(- \infty, 1) \cup (1, \infty)$

Step 10.2

Substitute any number, such as $0$ , from the interval $(- \infty, 1)$ in the first derivative $- 3 x^{2} + 6 x - 3$ to check if the result is negative or positive.

Tap for more steps...

Step 10.2.1

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

$f' (0) = - 3 {(0)}^{2} + 6 (0) - 3$

Step 10.2.2

Simplify the result.

Tap for more steps...

Step 10.2.2.1

Simplify each term.

Tap for more steps...

Step 10.2.2.1.1

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

$f' (0) = - 3 \cdot 0 + 6 (0) - 3$

Step 10.2.2.1.2

Multiply $- 3$ by $0$ .

$f' (0) = 0 + 6 (0) - 3$

Step 10.2.2.1.3

Multiply $6$ by $0$ .

$f' (0) = 0 + 0 - 3$

Step 10.2.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 10.2.2.2.1

Add $0$ and $0$ .

$f' (0) = 0 - 3$

Step 10.2.2.2.2

Subtract $3$ from $0$ .

$f' (0) = - 3$

Step 10.2.2.3

The final answer is $- 3$ .

$- 3$

Step 10.3

Substitute any number, such as $4$ , from the interval $(1, \infty)$ in the first derivative $- 3 x^{2} + 6 x - 3$ to check if the result is negative or positive.

Tap for more steps...

Step 10.3.1

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

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

Step 10.3.2

Simplify the result.

Tap for more steps...

Step 10.3.2.1

Simplify each term.

Tap for more steps...

Step 10.3.2.1.1

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

$f' (4) = - 3 \cdot 16 + 6 (4) - 3$

Step 10.3.2.1.2

Multiply $- 3$ by $16$ .

$f' (4) = - 48 + 6 (4) - 3$

Step 10.3.2.1.3

Multiply $6$ by $4$ .

$f' (4) = - 48 + 24 - 3$

Step 10.3.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 10.3.2.2.1

Add $- 48$ and $24$ .

$f' (4) = - 24 - 3$

Step 10.3.2.2.2

Subtract $3$ from $- 24$ .

$f' (4) = - 27$

Step 10.3.2.3

The final answer is $- 27$ .

$- 27$

Step 10.4

Since the first derivative did not change signs around $x = 1$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 10.5

No local maxima or minima found for $f (x) = - x^{3} + 3 x^{2} - 3 x$ .

No local maxima or minima

Step 11