Find the critical points of the function and use the First Derivative Test to de

Question

Find the critical points of the function and use the First Derivative Test to determine whether the critical point is a local minimum or maximum (or neither). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x) = 8x + e^(-7x) + 5

Local minimum    c = __________

Local maximum   c = __________

Determine the intervals on which the funciton is increasing or decreasing. (Enter your answers using interval notation. Enter EMPTY or 0/ for the empty set.)

Increasing __________

Decreasing _____________

Explanation / Answer

f(x) = 8x + e^(-7x) + 5

Deriving :

f'(x) = 8 - 7e^(-7x)

Now, f'(x) = 0 becomes :

8 - 7e^(-7x) = 0

e^(-7x) = 8/7

-7x = ln(8/7)

x = (-1/7)*ln(8/7) ---> This is the only critical point

Now, lets derive again to find f''(x)

f''(x) = 49e^(-7x) ---> Into this, plug n the critical number and we'd get a positive value

Since f'' is POSITIVE, it indicates that x = (-1/7)*ln(8/7) is a MINIMA.

When x = (-1/7)*ln(8/7), y = 8((-1/7)*ln(8/7)) + e^(-7*(-1/7)*ln(8/7)) + 5 --> y = 5.9902

So, local minimum = 5.9902 when x = -0.0191 ---> ANSWER
There is no local maximum ---> ANSWER

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Now, notice that x = (-1/7)*ln(8/7) is a MINIMUM.

So, this means, to the left of (-1/7)ln(8/7), it is decreasing and then to the right of (-1/7)ln(8/7), it is increasing.....

So, decreasing over (-inf , (-1/7)*ln(8/7)) ---> ANSWER
Increasing over ((-1/7)*ln(8/7) , inf) ---> ANSWER

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