Suppose that f(x)=(8-3x)e^x (A) List all the critical values of f(x). Note: If t

Question

Suppose that f(x)=(8-3x)e^x

(A) List all the critical values of f(x). Note: If there are no critical values, enter NONE

(B) Use interval notation to indicate where f(x) is increasing.
Increasing:

(d) Use interval notation to indicate where f(x) is decreasing.
Decreasing:

(D) List the x values of all local maxima of f(x). If there are no local maxima, enter NONE.
x values of local maximums =

(E) List the x values of all local minima of f(x). If there are no local minima, enter NONE.
x values of local minimums =

(F) Use interval notation to indicate where f(x) is concave up.
Concave up:

(G) Use interval notation to indicate where f(x) is concave down.
Concave down:

(H) List the x values of all the inflection points of f. If there are no inflection points, enter NONE.
x values of inflection points =

Explanation / Answer

f(x)=(8 - 3x)e^x

f'(x) = -3e^x + (8 - 3x)e^x
f'(x) = -3e^x + 8e^x - 3xe^x
f'(x) = 5e^x - 3xe^x
f'(x) = e^x(5 - 3x)....................(1)
f''(x) = e^x(5 - 3x) + (-3e^x)
f''(x) = 5e^x - 3xe^x - 3e^x
f''(x) = 2e^x - 3xe^x
f''(x) = e^x(2 - 3x).............(2)

(A)
for critical points, f'(x) = 0
f'(x) = e^x(5 - 3x) = 0
5 - 3x = 0
x = 5/3 is the critical point

(B)
for increasing, f'(x) > 0
f'(x) = e^x(5 - 3x) > 0
5 - 3x > 0
x < 5/3
x = (-infinity, 5/3).......increasing

(C)
for decreasing, f'(x) < 0
f'(x) = e^x(5 - 3x) < 0
5 - 3x < 0
x > 5/3
x = (5/3, infinity)........decreasing

(D)
f''(5/3) = e^x(2 - 3x) = e^5/3 (2 - 3*5/3) = -3e^5/3 < 0
Thus we have local maximum at x = 5/3
But we dont have any local minima
x values of local maximums =5/3

(E)
x values of local minimums = NONE

(F)
For concave up, f''(x) > 0
f''(x) = e^x(2 - 3x) > 0
2 - 3x > 0
x < 2/3
x = (-infinity, 2/3).........concave up

(G)
For concave down, f''(x) < 0
f''(x) = e^x(2 - 3x) < 0
2 - 3x < 0
x > 2/3
x = (2/3, infinity).........concave down

(H)
for inflection points, f''(x) = 0
e^x(2 - 3x) = 0
2 - 3x = 0
x = 2/3.........point of inflection

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