How do you determine whether the function

Willie Kelley

Willie Kelley

Answered question

2022-04-15

How do you determine whether the function f(x)=x2ex is concave up or concave down and its intervals?

Answer & Explanation

Kendall Clark

Kendall Clark

Beginner2022-04-16Added 8 answers

Step 1
f (x)=0
f(x)=x2×ex
f(x)=2×x×ex+x2×ex
f (x)=2×ex+2×x×ex+2×x×ex+x2×ex=
=ex(2+4×x+x2)
f (x)=02×ex(1+x+x2)=0
Remember that ex is always positive
(1+x+x2)=0
x1,2=b±b24ac2a
x1, 2=4±424×2×12×1=
=4±1682=
=4±82=
=4±222=
=2±2
x1=2+2=0.59
x2=22=3.41
now you have 3 intervals:
1. (, x2)
2. (x2, x1)
3. (x1, )
ask yourself if f (x) is positive or negative on each interval: (choose a number of the interval and calculate the value,don't choose x1 or x2 because there is 0 as we calculated before).
1. (, 3.41)x=4, f (4)>0

2. (3.41, 0.59)x=1, f (1)<0
3. (0.59, )x=0, f (0)>0
If f (x)>0f(x) is convex
If f (x)<0f(x) is concave
convex interval: (, 3.41)(0.59, )
concave interval: (3.41, 0.59)

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