1. Suppose that \( f(x)= 3x^4 -8x^3+6x^2 \).

(a) Find the intervals of increase or decrease.

(b) Find the local maximum and minimum values of \( f\).

Solution


(c) Find the intervals of concavity and the inflection points.

Solution


(d) Use the information from parts (a)-(c) to sketch the graph of \(f\).

Solution


2. Sketch the graph of a function that satisfies all of the following conditions.
i. Vertical asymptote at \( x=-2 \)
ii. \(\;f'(x) >0\) if \( -5 < x < -3 \), \( -2 < x < 0 \), or \( x >2 \)
iii. \(\;f'(x) <0\) if \( x < -5 \), \( -3 < x < -2 \), or \( 0< x <2 \)
iv. \(\;f''(x) >0\) if \( x < -4 \) or \( x >1 \)
v. \(\;f''(x) <0\) if \( -4 < x < -2 \) or \( -2< x <1\)

Solution


3. The graph of \(f' \) is given below. Find the following.


(a) The intervals where \( f \) is increasing or decreasing.
(b) The \(x\)-coordinate(s) of the local maximum and local minimum.
Solution
(c) The intervals where \(f\) is concave up or concave down.
(d) The \(x\)-coordinates of the inflection points.
Solution