Find increasing and decreasing intervals

Question

Consider the function 𝑓(𝑥)=𝑥2𝑒13𝑥.

For this function there are three important open intervals: (-∞,𝐴), (𝐴,𝐵), and (𝐵,∞) where 𝐴 and 𝐵 are the critical numbers.

Find 𝐴   

and 𝐵   

For each of the following intervals, tell whether 𝑓(𝑥) is increasing or decreasing.

(-∞,𝐴): Select an answer Increasing Decreasing 

(𝐴,𝐵): Select an answer Increasing Decreasing 

(𝐵,∞) Select an answer Increasing Decreasing 

Answer 1

I assume that your function is f(x) = x^2 * e^(13x) ?!

Then, f'(x) = 2x*e^(13x) + 13e^(13x) * x^2 = x*e^(13x)*(2+13x) = 0

CN: x= - 2/13, x=0

check f' values, so we know inc/dec [Using 1st derivative test]

f'(-1) = -2*e^(-13)+13*e^(-13)*1 = 11*e^-11 >0 =>inc

f'(-.0000001) =-.00000019999996 <0 => dec

f'(1) = 6636200.88 >0 => inc

Hence, (-inf, -2/13) , (-2/13, 0), (0,inf) are important intervals

Inc: (-inf, -2/13) U (0,inf)

Dec: (-2/13, 0)

Answer 2

Remember that critical numbers occur where the derivative is equal to zero and the derivative changes from positive to negative or vise versa. So you first need to find the derivative.

f'(x) = e^13x(13x² + 2x)

If you set this equal to zero, you should get two values of x = 0,-²/₁₃. Since the derivative aside from the exponential is a parabola opening up, it should be positive (increasing) before -²/₁₃ and after 0, leaving it negative (decreasing) between those values