How do i answer this?

Question

For the function 𝑔(𝑥) = 𝑥𝑒 ^−2𝑥 find all critical points and say which are local maximums or minimums. Find the point of inflection and say where the function is concave up and where it is concave down.

Answer 1

To find the critical points, we need to take the derivative and set it equal to 0. we have:

f'(x) = e^-2x - 2xe^-2x = 0

e^-2x(1 - 2x) = 0

x = 1/2

Since f'(x) > 0 for x < 1/2 and f'(x) < 0 for x > 1/2, the point (1/2, 1/(2e)) is a local maximum. Now, we take the derivative again and set it equal to 0 to find the points of inflection. We have:

f''(x) = -2e^-2x - 2e^-2x + 4xe^-2x = -4e^-2x + 4xe^-2x = 0

4e^-2x(x - 1) = 0

x = 1

The point of inflection is therefore (1, e^-2). Since f''(x) > 0 for x > 1, f is concave up for (1, ∞), and since f''(x) < 0 for x < 1, f is concave down for (-∞, 1).

Answer 2

g(x) = ex - 2x

g(x) = (e - 2)x

Straight line with slope of e - 2, x and y intercept of (0, 0).