6.4 Mean-Value Type Theorems 327
Prove that the equation 3x^4 - 8x^3 + 7x^2 - 45 = 0 has exactly two real
roots. [You must prove two things: that the equation has at least two real
roots, and that it cannot have more than two.]
Suppose f"(x) = 0 for all x ER Prove that f must be a polynomial and
have degree :::; 1.
Suppose f"'(x) = 0 for all x ER Prove that f must be a polynomial and
have degree :::; 2.
Suppose f: IR-+ IR, and 'Vx,y E IR, lf(x)-f(y)I :S: lx-yl^2 • Prove that f
is a constant function.
Suppose f is differentiable everywhere, f (-1) = 5, f(O) = 0, and f(l) =
Prove that :Jc, d E (-1, 1) 3 f'(c) = -3 and f'(d) = 3. [Hint: You
will find both the MVT and Theorem 6.3.7 helpful.]
Prove Theorem 6.4.6 (b).
Prove Theorem 6.4.6 (c) and (d).
Prove by examples that the converses of Theorem 6.4.6 (c) and (d) are
false.
Use the mean va1ue theorem to prove that 'Vx, y E IR, I cos x - cos YI :::;
lx-yl.
Use the mean value theorem to prove that 'Vx E (0, ~), tanx > x.
Use the mean value theorem to prove that I;/ 0 < x < y,
y-x y y-x
--<ln-<--.
y x x
x-1
- Use the mean value theorem to prove that 'Vx > 1, --< ln x < x - 1.
x - (a) Prove that the function f(x) = sinx is strictly decreasing on (o, ~].
x
[Hint: Use Exercise 17 to show that f'(x) < 0 on (0, ~] .]
2x
(b) Use the result of (a) to prove that sinx > - on (0, ~].
7f - Suppose f' is continuous at some interior point xo of its domain. Prove
that
(a) if f'(xo) > 0, f is strictly increasing on some neighborhood of xo.
(b) if f' ( x 0 ) < 0, f is strictly decreasing on some neighborhood of xo.