SAMPLE TEST # 3
Show your work.
1. Let f(x) = x4−6x2be on the interval [−2,3].
(a) Find all points at which fhas local maximum or a local minimum. Show your work.
(b) Find all points at which fhas an inflection point. Show your work.
(c) Graph fusing (a) and (b). Mark on the graph: all local extrema, inflection points,
concavity, and x, y-intercepts.
2. Assume that f(3) = 7 and that f0(x)≥5 for all values of x. Use Mean Value Theorem
to determine what is the smallest possible value of f(6).
3. Find the limits.
(a) lim
x→−∞ ³x+√x2+ 2x´
(b) lim
x→∞
√x2−9
2x−6
4. Find the horizontal and vertical asymptotes of the function f(x) = x3+x2−1
2x3+x.
5. Find the point on the line 2x+ 3y+ 5 = 0 that is closest to the point (−1,−2).
6. The sum of two nonnegative numbers is 10. Find the minimum possible value of the sum
of their cubes.
7. Sketch the graph of the function that satisfies all of the given conditions.
f0(2) = 0, f0(0) = 1,
f0(x)>0 if 0 < x < 2, f0(x)<0 if x > 2,
f00(x)<0 if 0 < x < 4, f00 (x)>0 if x > 4,
limx→∞ f(x) = 0,
f(−x) = −f(x) for all x.
8. Use the second derivative test to find local maxima and local minima of the function
f(x) = x3−3x+ 1.
9. Find absolute maximum and absolute minimum of f(x) = 2x3+x2−x+ 1 on [0,2].
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