Exercise 10

For a certain function $f$, its derivative is known to be $f'(x) = (x-1)e^{-x^2}$. Note that you do not know a formula for $y = f(x)$.

1. At what $x$-value(s) is $f'(x) = 0$? Justify your answer algebraically, but include a graph of $f'$ to support your conclusion.

2. Reasoning graphically, for what intervals of $x$-values is $f''(x) \gt 0$? What does this tell you about the behavior of the original function $f$? Explain.

3. Assuming that $f(2) = -3$, estimate the value of $f(1.88)$ by finding and using the tangent line approximation to $f$ at $x=2$. Is your estimate larger or smaller than the true value of $f(1.88)$? Justify your answer.