009A Sample Final 3, Problem 6

Let

$f(x)=4+8x^{3}-x^{4}$

(a) Over what  $x$-intervals is  $f$  increasing/decreasing?

(b) Find all critical points of  $f$  and test each for local maximum and local minimum.

(c) Over what  $x$-intervals is  $f$  concave up/down?

(d) Sketch the shape of the graph of  $f.$

Foundations:
1. $f(x)$  is increasing when  $f'(x)>0$  and  $f(x)$  is decreasing when  $f'(x)<0.$
2. The First Derivative Test tells us when we have a local maximum or local minimum.
3. $f(x)$  is concave up when  $f''(x)>0$  and  $f(x)$  is concave down when  $f''(x)<0.$

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

(c)

Step 1:

Step 2:

(d):
Insert graph

Final Answer:
(a)
(b)
(c)
(d)    See above

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