How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x) = x^2e^x - 3#?

Question

Samantha Adkison · Accepted Answer

#f(x)# is strictly increasing in #(-oo,-2)#, has a local maximum in #x=-2#, it is strictly decreasing in #(-2,0)# reaching a local minimum in #x=0# and again increasing for #x in (0,+oo)#

Given:

#f(x) = x^2e^x-3#

consider the derivative of the function:

#f'(x) = d/dx (x^2e^x-3) = (d/dx x^2)e^x + x^2 (d/dx e^x)#

#f'(x) = 2xe^x+x^2e^x#

#f'(x) = x(2+x)e^x#

Solve now the inequality:

#f'(x) > 0#

As #e^x > 0 AA x in RR#, the sign of #f'(x)# is the sign of #x(2+x)#, then we have:

#f'(x) > 0# for #x in (-oo,-2) uu (0,+oo)#

#f'(x) < 0# for #x in (-2,0)#

and the critical points where #f'(x) = 0# are #x_1 =-2# and #x_2 =0#

We can conclude that #f(x)# is strictly increasing in #(-oo,-2)#, has a local maximum in #x=-2#, it is strictly decreasing in #(-2,0)# reaching a local minimum in #x=0# and again increasing for #x in (0,+oo)#

graph{x^2e^x-3 [-10, 10, -5, 5]}

Cameron Alexander · Answer

undefined To determine where the function is increasing or decreasing, and to find relative maxima and minima for $ f(x) = x^2e^x - 3 $, follow these steps: 1. Calculate the first derivative of the function $ f'(x) $. 2. Set $ f'(x) = 0 $ and solve for $ x $. These are the critical points. 3. Determine the intervals where $ f'(x) $ is positive (increasing) and where it is negative (decreasing). 4. Check the behavior of the function around the critical points to identify relative maxima and minima. Let's go through each step: 1. The first derivative of $ f(x) $ is $ f'(x) = 2xe^x + x^2e^x $. 2. Setting $ f'(x) = 0 $, we get: $ 2xe^x + x^2e^x = 0 $ Factoring out $ e^x $, we have: $ e^x(2x + x^2) = 0 $ This equation is true when either $ e^x = 0 $ (which is not possible) or when $ 2x + x^2 = 0 $. Solving $ 2x + x^2 = 0 $, we find critical points at $ x = 0 $ and $ x = -2 $. 3. We need to test the intervals around these critical points to determine where the function is increasing or decreasing. We can use test points from each interval and evaluate $ f'(x) $: - For $ x < -2 $: Pick $ x = -3 $, $ f'(-3) > 0 $. So, the function is increasing. - For $ -2 < x < 0 $: Pick $ x = -1 $, $ f'(-1) < 0 $. So, the function is decreasing. - For $ x > 0 $: Pick $ x = 1 $, $ f'(1) > 0 $. So, the function is increasing. 4. Now, to find relative maxima and minima: - At $ x = -2 $, the function changes from decreasing to increasing, indicating a local minimum. - At $ x = 0 $, the function changes from increasing to decreasing, indicating a local maximum. Therefore, the function $ f(x) = x^2e^x - 3 $ has a relative minimum at $ x = -2 $ and a relative maximum at $ x = 0 $.