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14. For each of the following graphs of the function y f 1x2, make a rough
sketch of the derivative function f ¿1x2. By comparing the graphs of f 1x2
and f ¿1x2, show that the intervals for which f 1x2 is increasing correspond
to the intervals where f ¿1x2 is positive. Also show that the intervals where
f 1x2 is decreasing correspond to the intervals for which f ¿1x2 is negative.
a. y
c.
y
4
4
y = f(x)
2 y = f(x)
2
x
x
–4 –2 0 2 4
–2 –1 0 1 2 3
–2
–2
f 1x2 is a linear function.
b. y
d.
4
y = f(x)
2
x
–4 –2 0 2 4
–2
f 1x2 is a cubic function.
4
y
2 y = f(x)
x
–4 –2 0
–2
2 4
f 1x2 is a quadratic function. f 1x2 is a quartic function.
15. Consider the function f 1x2 3x 4 ax 3 bx 2 cx d.
a. Find constants a, b, c, and d such that the graph of f will have horizontal
tangents at 12, 732 and 10, 92.
b. There is a third point that has a horizontal tangent. Find this point.
c. For all three points, determine whether each corresponds to a local
maximum, a local minimum, or neither.
PART C
16. For each of the following polynomials, find the local extrema and the
direction that the curve is opening for x 100. Use this information to make
a quick sketch of the curve.
a. y 4 3x 2 x 4
b. y 3x 5 5x 3 30x
17. Suppose that f1x2 and g1x2 are positive functions (functions where f 1x2 7 0
and g1x2 7 0) such that f1x2 has a local maximum and g1x2 has a local
minimum at x c. Show that the function h1x2 f 1x2 has a local maximum
g1x2
at x c.
180 4.2 CRITICAL POINTS, LOCAL MAXIMA, AND LOCAL MINIMA
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