Calculus Examples

$f (x) = 5 x - 2 x^{3} - 2$ is concave down at $x = 1$

Step 1

Find the corresponding $y$ -value to $x = 1$ .

Tap for more steps...

Step 1.1

Substitute $1$ in for $x$ .

$y = 5 (1) - 2 {(1)}^{3} - 2$

Step 1.2

Solve for $y$ .

Tap for more steps...

Step 1.2.1

Remove parentheses.

$y = 5 (1) - 2 \cdot 1^{3} - 2$

Step 1.2.2

Remove parentheses.

$y = 5 (1) - 2 {(1)}^{3} - 2$

Step 1.2.3

Simplify $5 (1) - 2 {(1)}^{3} - 2$ .

Tap for more steps...

Step 1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.2.3.1.1

Multiply $5$ by $1$ .

$y = 5 - 2 {(1)}^{3} - 2$

Step 1.2.3.1.2

One to any power is one.

$y = 5 - 2 \cdot 1 - 2$

Step 1.2.3.1.3

Multiply $- 2$ by $1$ .

$y = 5 - 2 - 2$

Step 1.2.3.2

Simplify by subtracting numbers.

Tap for more steps...

Step 1.2.3.2.1

Subtract $2$ from $5$ .

$y = 3 - 2$

Step 1.2.3.2.2

Subtract $2$ from $3$ .

$y = 1$

Step 2

Find the first derivative and evaluate at $x = 1$ and $y = 1$ to find the slope of the tangent line.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $5 x - 2 x^{3} - 2$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 2]$ .

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $5$ by $1$ .

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $3$ by $- 2$ .

$5 - 6 x^{2} + \frac{d}{d x} [- 2]$

Step 2.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$5 - 6 x^{2} + 0$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Add $5 - 6 x^{2}$ and $0$ .

$5 - 6 x^{2}$

Step 2.5.2

Reorder terms.

$- 6 x^{2} + 5$

Step 2.6

Evaluate the derivative at $x = 1$ .

$- 6 {(1)}^{2} + 5$

Step 2.7

Simplify.

Tap for more steps...

Step 2.7.1

Simplify each term.

Tap for more steps...

Step 2.7.1.1

One to any power is one.

$- 6 \cdot 1 + 5$

Step 2.7.1.2

Multiply $- 6$ by $1$ .

$- 6 + 5$

Step 2.7.2

Add $- 6$ and $5$ .

$- 1$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

Tap for more steps...

Step 3.1

Use the slope $- 1$ and a given point $(1, 1)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - 1 \cdot (x - (1))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y - 1 = - 1 \cdot (x - 1)$

Step 3.3

Solve for $y$ .

Tap for more steps...

Step 3.3.1

Simplify $- 1 \cdot (x - 1)$ .

Tap for more steps...

Step 3.3.1.1

Rewrite.

$y - 1 = 0 + 0 - 1 \cdot (x - 1)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 1 = - 1 \cdot (x - 1)$

Step 3.3.1.3

Apply the distributive property.

$y - 1 = - 1 x - 1 \cdot - 1$

Step 3.3.1.4

Simplify the expression.

Tap for more steps...

Step 3.3.1.4.1

Rewrite $- 1 x$ as $- x$ .

$y - 1 = - x - 1 \cdot - 1$

Step 3.3.1.4.2

Multiply $- 1$ by $- 1$ .

$y - 1 = - x + 1$

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 3.3.2.1

Add $1$ to both sides of the equation.

$y = - x + 1 + 1$

Step 3.3.2.2

Add $1$ and $1$ .

$y = - x + 2$

Step 4