Calculus Examples

$f (x) = 3 t^{5} - 15 t^{4} - 40 t^{3} + 80$

Step 1

Find the second 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 $3 t^{5} - 15 t^{4} - 40 t^{3} + 80$ with respect to $x$ is $\frac{d}{d x} [3 t^{5}] + \frac{d}{d x} [- 15 t^{4}] + \frac{d}{d x} [- 40 t^{3}] + \frac{d}{d x} [80]$ .

$\frac{d}{d x} [3 t^{5}] + \frac{d}{d x} [- 15 t^{4}] + \frac{d}{d x} [- 40 t^{3}] + \frac{d}{d x} [80]$

Step 1.1.2

Since $3 t^{5}$ is constant with respect to $x$ , the derivative of $3 t^{5}$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 15 t^{4}] + \frac{d}{d x} [- 40 t^{3}] + \frac{d}{d x} [80]$

Step 1.1.3

Since $- 15 t^{4}$ is constant with respect to $x$ , the derivative of $- 15 t^{4}$ with respect to $x$ is $0$ .

$0 + 0 + \frac{d}{d x} [- 40 t^{3}] + \frac{d}{d x} [80]$

Step 1.1.4

Since $- 40 t^{3}$ is constant with respect to $x$ , the derivative of $- 40 t^{3}$ with respect to $x$ is $0$ .

$0 + 0 + 0 + \frac{d}{d x} [80]$

Step 1.1.5

Since $80$ is constant with respect to $x$ , the derivative of $80$ with respect to $x$ is $0$ .

$0 + 0 + 0 + 0$

Step 1.1.6

Combine terms.

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Step 1.1.6.1

Add $0$ and $0$ .

$0 + 0 + 0$

Step 1.1.6.2

Add $0$ and $0$ .

$0 + 0$

Step 1.1.6.3

Add $0$ and $0$ .

$f' (x) = 0$

Step 1.2

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

$f'' (x) = 0$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $0$ .

$0$

Step 2

Set the second derivative equal to $0$ then solve the equation $0 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$0 = 0$

Step 2.2

Since $0 = 0$ , the equation will always be true.

Always true

Step 3

There are no inflection points in a straight line, $f (x) = 3 t^{5} - 15 t^{4} - 40 t^{3} + 80$ .

No Inflection Points