# Consider the following.f(x) = −5x2 + 40xFind the critical numbers. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)x =     Find the open intervals on which the function is increasing or decreasing. Use a graphing utility to verify your results. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)increasing       decreasing

Question

Consider the following.

f(x) = −5x2 + 40x

Find the critical numbers. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

x =

Find the open intervals on which the function is increasing or decreasing. Use a graphing utility to verify your results. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)

 increasing decreasing
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Step 1

f(x) = −5x2 + 40x

Hence, f'(x) = -10x + 40

Step 2

The critical values of x are those values of x at which f'(x) is zero or it doesn't exist.

f'(x) = -10x + 40 = 0

Hence, x = 40/10 = 4

f'(x) exists at all the points and values of x as it's a polynomial function.

Hence, the crtical number is:

x = 4

Step 3

If f′(x) > 0 at each point in an interval I, then the function is said to be increasing on I.

If &...

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