Precalculus Examples

$H (T) = - 16 t^{2} + 80 t + 6$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $- 16 t^{2} + 80 t + 6$ with respect to $T$ is $\frac{d}{d T} [- 16 t^{2}] + \frac{d}{d T} [80 t] + \frac{d}{d T} [6]$ .

$\frac{d}{d T} [- 16 t^{2}] + \frac{d}{d T} [80 t] + \frac{d}{d T} [6]$

Step 1.2

Since $- 16 t^{2}$ is constant with respect to $T$ , the derivative of $- 16 t^{2}$ with respect to $T$ is $0$ .

$0 + \frac{d}{d T} [80 t] + \frac{d}{d T} [6]$

Step 1.3

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

$0 + 0 + \frac{d}{d T} [6]$

Step 1.4

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

$0 + 0 + 0$

Step 1.5

Combine terms.

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Step 1.5.1

Add $0$ and $0$ .

$0 + 0$

Step 1.5.2

Add $0$ and $0$ .

$0$

Step 2

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

$H'' (T) = 0$

Step 3

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

$0 = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema