320 Chapter 6 • Differentiable Functions
Thus, by (a) and (b), g must have its maximum value for [a, b] at some
point c E (a, b). Then g has a local maximum at c. Hence, by Theorem 6.3.4,
g'(c) = 0.
That is,
k-f'(c)=O;
f'(c) = k.
Case 2 (f'(a) > k > f'(b)): Exercise 9. •
EXERCISE SET 6.3l. Prove Theorem 6.3.3.
- Prove Case 2 of Theorem 6.3.4.
- Prove Theorem 6.3.5 (b).
- Suppose both functions f, g : JR--> JR have local maxima at xa. Which of
the following must be true? [Give a proof or a counterexample.]
(a) f + g has a local maximum at xa.
(b) f g has a local maximum at x 0. - Use Theorem 6.3.5 and its converse (not yet proved) to find the interval(s)
over which the given function is increasing, and the interval(s) over which
it is decreasing. Also, find the local extreme values of each function:
(a) f(x)=lx^2 -x-61
( c) f ( x) = x2 ~ 1
( e) f ( x) = x2 ~ 1
1
(b) f(x) = x + -
x
x
(d) f(x) = x2 - 1
x
(f) f(x)=x2+1- Give an example of a function f : JR --> JR, which is strictly increasing and
yet f' ( x) is not everywhere > 0. - Let f be the function defined in Example 6.3.6. Prove:
(a) f is differentiable everywhere, and f'(O) > 0.
(b) f is not monotone in any neighborhood of 0. [Show that every neigh-
borhood of 0 must contain a tail of the sequence { nln} and apply The-
orem 6.3.5.] - Prove that for the function f of Exercise 7, f' is not continuous at 0.
- Complete the proof of Theorem 6.3.7 by proving Case 2.