Mean Value Theorem

Theorem 3.2.1

If ff is differentiable on an open interval (a,b)(a,b) and ff assumes its maximum or minimum at a point c(a,b)c \in (a,b), then f(c)=0f^\prime(c) = 0.

Suppose that ff assumes its maximum at cc. That is, f(x)f(c)f(x) \leq f(c) for all x(a,b)x \in (a,b). Let (xn)(x_n) be a sequence converging to cc such that a<xn<ca < x_n < c for all nn. Since ff is differentiable at cc, the sequence

(f(xn)f(c)xnc) \left( \frac{f(x_n) - f(c)}{x_n - c} \right)

converges to f(c)f^\prime(c). Since f(xn)f(c)f(x_n) \leq f(c) and xncx_n \leq c, each term in our above sequence is nonnegative. Thus f(c)0f^\prime(c) \geq 0.

We repeat this argument with a sequence (yn)(y_n), such that c<yn<bc < y_n < b for all nn. Each term in the sequence

(f(yn)f(c)ync) \left( \frac{f(y_n) - f(c)}{y_n - c} \right)

will be nonpositive, so f(c)0f^\prime(c) \leq 0. We therefore conclude that f(c)=0f^\prime(c) = 0. If ff has a minimum at cc, we apply the baove results to the function f-f.

Theorem 3.2.2 (Rolle’s Theorem)

Let ff be a continuous function on [a,b][a,b] that is differentiable on (a,b)(a,b) and such that f(a)=f(b)f(a) = f(b). Then there exists at least one point cc in (a,b)(a,b) such that f(c)=0f^\prime(c) = 0.

Since ff is continuous on a compact interval [a,b][a,b], ff obtains a minumum x1x_1 and maximum x2x_2 by the Extreme Value Theorem.

If x1,x2x_1,x_2 are both endpoints of [a,b][a,b], then the function is a constant, and f(x)=0f^\prime(x) = 0 for all x[a,b]x \in [a,b].

Otherwise, ff assumes either a minimum or maximum at some point c(a,b)c \in (a,b), and by Theorem 3.2.13.2.1, f(c)=0f^\prime(c) = 0.

Theorem 3.2.3 (Mean Value Theorem)

Let ff be a continuous function on [a,b][a,b] that is differentiable on (a,b)(a,b). Then there exists at least one point c(a,b)c \in (a,b) such that

f(c)=f(b)f(a)ba. f^\prime(c) = \frac{f(b) - f(a)}{b-a}.

Let g(x)g(x) be a function whose graph is the chord between f(a)f(a) and f(b)f(b). More formally

g(x)=f(b)f(a)ba(xa)+f(a),for all x[a,b]. g(x) = \frac{f(b)-f(a)}{b-a}(x-a) + f(a), \quad \text{for all $x \in [a,b]$}.

Then the function h=fgh = f - g is continuous on [a,b][a,b] and differentiable on (a,b)(a,b). Since f(a)=g(a)f(a) = g(a) and f(b)=g(b)f(b) = g(b), we have that h(a)=h(b)=0h(a) = h(b) = 0. Applying Rolle’s Theorem we see that for some c(a,b)c \in (a,b),

0=h(c)=f(c)g(c)=f(c)f(b)f(a)ba. 0 = h^\prime(c) = f^\prime(c) - g^\prime(c) = f^\prime(c) - \frac{f(b) - f(a)}{b-a}.

Thus

f(c)=f(b)f(a)ba. f^\prime(c) = \frac{f(b)-f(a)}{b-a}.

Theorem 3.2.4

Let ff be a continuous function on [a,b][a,b] that is differentiable on (a,b)(a,b). If f(x)=0f^\prime(x) = 0 for all x(a,b)x \in (a,b), then ff is constant on [a,b][a,b].

Proof by Contradiction

Suppose ff were not constant on [a,b][a,b]. Then there exists two points ax1<x2ba \leq x_1 < x_2 \leq b such that f(x1)f(x2)f(x_1) \not = f(x_2). But then by MVT, for some c(x1,x2)c \in (x_1,x_2) there exists

f(c)=f(x2)f(x1)x2x10 f^\prime(c) = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \not = 0

Thus a contradiction.

Corollary 3.2.5

Let f,gf,g be continuous on [a,b][a,b] and differentiable on (a,b)(a,b). Suppose that f(x)=g(x)f^\prime(x) = g^\prime(x) for all x(a,b)x \in (a,b). Then there exists a constant CC such that

f(x)=g(x)+C for all x[a,b] f(x) = g(x) + C \quad \text{ for all $x \in [a,b]$}

Direct Proof

Apply Theorem 3.2.43.2.4 to g-g.

Theorem 3.2.6

Let ff be differentiable on II. Then

(a)(a) if f(x)>0f^\prime(x) > 0 for all xIx \in I, then ff is strictly increasing on II, and

(b)(b) if f(x)<0f^\prime(x) < 0 for all xIx \in I, then ff is strictly decreasing on II.

Direct Proof

Theorem 3.2.7 (Intermediate Value Theorem for Derivatives)

Let ff be differentiable on [a,b][a,b] and suppose there exists a number kk between f(a)f^\prime(a) and f(b)f^\prime(b). Then there exists a point c(a,b)c \in (a,b) such that f(c)=kf^\prime(c) = k.

(a)(a) if f(x)>0f^\prime(x) > 0 for all xIx \in I, then ff is strictly increasing on II, and

(b)(b) if f(x)<0f^\prime(x) < 0 for all xIx \in I, then ff is strictly decreasing on II.

Direct Proof