Problem 19. (18 points)
Suppose that $f(x) = x^4 - 6x^3$.
(A) List all the critical values of $f(x)$. Note: If there are no critical values, enter 'NONE'.
(B) Use interval notation to indicate where $f(x)$ is increasing.
Note: Use 'INF' for $\infty$, '-INF' for $-\infty$, and use 'U' for the union symbol.
Increasing:
(C) Use interval notation to indicate where $f(x)$ is decreasing.
Decreasing:
(D) List the $x$ values of all local maxima of $f(x)$. If there are no local maxima, enter 'NONE'.
x values of local maximums =
(E) List the $x$ values of all local minima of $f(x)$. If there are no local minima, enter 'NONE'.
x values of local minimums = (F) Use interval notation to indicate where $f(x)$ is concave up.
Concave up:
(G) Use interval notation to indicate where $f(x)$ is concave down.
Concave down:
(H) List the $x$ values of all the inflection points of $f$. If there are no inflection points, enter 'NONE'.
x values of inflection points =
(I) Use all of the preceding information to sketch a graph of $f$. When you're finished, enter a "1" in the box below.
Note: You can earn partial credit on this problem.
