Review Math Notes box in this lesson. Let $f(x)=-\frac{1}{4}x^4+3x^3-10x^2+40$. . ​

1. Identify the local maxima and local minima of $f$.

   Step 1(a):

   Find CANDIDATES for maxima and minima by setting the first derivative equal to $0$.

   Step 2(a):

   Decide if each candidate is a maximum, minimum, or neither. You can check for slope change by evaluating the first derivative at points close to each candidate. Or, you can evaluate each candidate in the second derivative. If the graph is concave up, it cannot have a local max, and vice versa.

   Step 3(a):

   Recall that maxima and minima are $y$-values, not $x$-values. So use $f\left(x\right)$ to determine the $y$-value of each candidate.

2. Identify the global maximum and global minimum of $f$.

   Hint (b):

   Graph $y = f\left(x\right)$. Which points are global extrema?

3. Identify the global maximum and global minimum values of $f$ over the interval $[2, 5]$.

   Hint (c):

   Be sure to evaluate the function at the endpoints of the interval. These points are candidates for extrema.

4. Identify the global maximum and global minimum values of $f$ over the interval $[–2, 1]$.

5. Explain how your answers to parts (a) through (d) demonstrate the Extreme Value Theorem.

   Hint (e):

   Do global extrema always exist? If so, when? If not, why not?