Calc 12
posted by Isaac
Let f be the function defined by f(x) = (x^2 + 1)e^x for 4≤x≤4.
a. For what value of x does f reach its absolute maximum? Justify your answer. b. Find the xcoordinates of all points of inflection of f. Justify your answer.

Reiny
by the product rule ...
f '(x) = (x^2 + 1)(1)e^(x) + 2x e^x
= e^x (x^2  1 + 2x)
for a max or min, f '(x) = 0
e^x (x^2  1 + 2x)
e^x = 0 > not possible
or
x^2 2x + 1 = 0
(x1)^2 =
x = 1
f(1) = (1+1)e^1= 2/e = appr .736
check the ends since we are given a domain:
f(4) = 17*e^4 = appr 928
f(4) = 17/e^4 = appr .311
So x = 4 yields the maximum
while x=1 would yield a local maximum
points of inflection f ''(x) = 0
f ''(x) = (x1)^2 (e^x) + 2(x1)e^x
= e^x((x1)^2 + 2(x1)
= e^x (x^2 + 4x 3)
= 0
x^2  4x + 3 = 0
(x1)(x3) = 0
points of inflection at x = 1 and x = 3
observe (1, 2/e) is both a turning point and a point of inflection
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