http://www.ck12.org Chapter 3. Applications of Derivatives
Proof:Consider the graph offand secant linesas indicated in the figure.
By the Point-Slope form of lineswe have
y−f(a) =m(x−a)andy=m(x−a)+f(a).
For eachxin the interval(a,b),letg(x)be the vertical distance from lineSto the graph off.Then we have
g(x) =f(x)−[m(x−a)+f(a)]for everyxin(a,b).
Note thatg(a) =g(b) = 0 .Sincegis continuous in[a,b]andg′exists in(a,b),then Rolle’s Theorem applies. Hence
there existscin(a,b)withg′(c) = 0.
Sog′(x) =f′(x)−mfor everyxin(a,b).
In particular,
g′(c) =f′(c)−m=0 and
f′(c) =m
f′(c) =f(bb)−−af(a)
f(b)−f(a) = (b−a)f′(c).The proof is complete.
Example 2:
Verify that the Mean Value Theorem applies for the functionf(x) =x^3 + 3 x^2 − 24 xon the interval[ 1 , 4 ].
Solution:
We need to findcin the interval( 1 , 4 )such thatf( 4 )−f( 1 ) = ( 4 − 1 )f′(c).
Note thatf′(x) = 3 x^2 + 6 x− 24 ,andf( 4 ) = 16 ,f( 1 ) =− 20 .Hence, we must solve the following equation:
36 = 3 f′(c)
12 =f′(c).By substitution, we have
12 = 3 c^2 + 6 c− 24
3 c^2 + 6 c− 36 = 0
c^2 + 2 c− 12 = 0
c=−^2 ±