Question

Let f(x)=e^(x^(2)+2x) Where does f have critical points? Choose all answers that apply: A x=-2 B x=-1 C x=0 D f has no critical points.

Answer 1

To find the critical points of the function, we first need to take the derivative of the function.

The given function is f(x) = e^(x^2 + 2x).

We apply the chain rule to differentiate the function, as it is a composition of two functions—the exponential function and a quadratic function.

The derivative of an exponential function is the function itself, and for the quadratic function x^2 + 2x, its derivative is 2x + 2. By applying the chain rule, we multiply these two derivatives.

Hence, the derivative f'(x) is e^(x^2 + 2x) *(2x + 2).

The definition of a critical point is that the derivative of the function either equals zero or undefined.

So, let's solve for when the derivative equals to zero. Since the factor e^(x^2 + 2x) is never zero, the only solution to f'(x) = 0 is when the second factor is zero.

Now we solve the equation 2x + 2 = 0. Solve for x, we have x = -1.

Hence, the function f(x) = e^(x^2 + 2x) has one critical point at x = -1. Therefore, the correct answer is B: x = -1.