Mir A. answered • 11/10/15

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Your derivative is correct.

(b) Critical points are those points on the graph of f(x) where f'(x) is zero. So set the first derivative equal to 0 and solve for x:

- 9 e

^{-9x}x^{3}+ 3 e^{-9x}x^{2}= 0Since e

^{-9x}can never be equal to zero, the only way this equation would be true is if:- 9 x

^{3}+ 3 x^{2}= 0x

^{2}(3 - 9x) = 0x = 0 or x = 1/3

So these are the critical numbers.

(c) We divide the entire number line in three parts based on the two critical points:

-inf < x < 0

0 < x < 1/3

1.3 < x < inf

inf stands for infinity.

Now just pick a number from each of these intervals and plug in to the first derivative and just note the sign of the derivative which says whether the function is decreasing or increasing:

-inf < x < 0 sign of f'(x) = +, thus function is increasing

0 < x < 1/3 sign of f'(x) = +, thus function is increasing

1.3 < x < inf sign of f'(x) = -, thus function is decreasing

0 < x < 1/3 sign of f'(x) = +, thus function is increasing

1.3 < x < inf sign of f'(x) = -, thus function is decreasing

Mir A.

Sure, my pleasure. If you need tutoring on Calculus, let me know, I can help through the Wyzant online tutoring platform.

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11/10/15

Kimberly K.

11/10/15