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.

Calc 12 
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
Respond to this Question
Similar Questions

calc
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? 
New Age Math
Suppose that y≤5x,3x≤y and 14x+15y≤1 together with 0≤x, 0≤y. The maximum value of the function y+x on the resulting region occurs at x=? 
calculus
4. Given the function f defined by f(x) = cos2x for π≤ x ≤π a. Find the xintercepts of the graph of f. b. Find the x and y coordinates of all relative maximum points of f. Justify your answer. c. Find the intervals … 
calculus
1. Consider the function f(x) = X^(4/3) +4x^(1/3) on the interval 8 ≤ x ≤ 8. a. Find the coordinate of all points at which the tangent to the curve is a horizontal line. b. Find the coordinate s of all points at which … 
calculus
1. Consider the function f(x) = X^(4/3) +4x^(1/3) on the interval 8 ≤ x ≤ 8. a. Find the coordinate of all points at which the tangent to the curve is a horizontal line. b. Find the coordinate s of all points at which … 
Calculus
If m ≤ f(x) ≤ M for a ≤ x ≤ b, where m is the absolute minimum and M is the absolute maximum of f on the interval [a, b], then m(b − a) ≤ b f(x) dx a≤ M(b − a). Use this property to estimate … 
Calculus
If m ≤ f(x) ≤ M for a ≤ x ≤ b, where m is the absolute minimum and M is the absolute maximum of f on the interval [a, b], then m(b − a) ≤ b f(x) dx a≤ M(b − a). Use this property to estimate … 
PRE  CALCULUS
Eliminate the parameter t. Find a rectangular equation for the plane curve defined by the parametric equations. x = 6 cos t, y = 6 sin t; 0 ≤ t ≤ 2π A. x2  y2 = 6; 6 ≤ x ≤ 6 B. x2  y2 = 36; 6 ≤ … 
Calculus
f is a continuous function with a domain [−3, 9] such that f(x)= 3 , 3 ≤ x < 0 x+3 , 0 ≤ x ≤ 6 3 , 6 < x ≤ 9 and let g(x)= ∫ f(t) dt where a=2 b=x On what interval is g increasing? 
Calculus
Let f be a differentiable function defined on the closed interval [a,b] and let c be a point in the open interval (a,b) such that I.f'(c)=0 II.f'(x)>0 when a≤x<c III.f'(x)<0 when c<x<≤b Which of the following …