Question

Suppose that g ' (x) = xf '(x ^2 ), where f (0) = -1, f (1) = 2,and g (0) = 0. What is g (-1)?

Answer 1

To find g(-1), evaluate g'(x) = xf'(x^2) at x = -1.

Step-by-step explanation:

To find g(-1), we need to evaluate the expression g'(x) = xf'(x^2) at x = -1. Given f(0) = -1 and f(1) = 2, we can find f' using the power rule for derivatives. Since f(x) is not given explicitly in the question, we will assume it is a polynomial function. Evaluating the derivative at x^2, we have:

f'(x^2) = 2x^2 - 1

Now, substituting x = -1, we get:

f'((-1)^2) = 2(-1)^2 - 1 = 2 - 1 = 1

Finally, substituting this result into the expression for g'(x), we have:

g'(-1) = (-1)(1) = -1