推 ERT312 : f'(c)不一定會等於k 11/05 22:48
※ 引述《niceperson (一條香蕉)》之銘言：
: suppose that function f(x) is continuous on [a,b]and differentiable on (a,b),and 0<a<b.
: If. f(a)=ka
: f(b)=kb. for some k
: show that there exists c∈(a,b) s.t. the tangent line of y=f(x) at c passes through the origin
: 不太會解這個
: 煩請各位大大幫忙
: 手機排版請見諒
: -----
: Sent from JPTT on my InFocus M330.
f(x)
Proof: Let g(x) = ------ Then g(a) = g(b) = k
x
Since f(x) is continuous on [a,b] and differentiable on (a,b) ,
g(x) is continuous on [a,b] and differentiable on (a,b)
By Rolle's theorem , there exists c∈(a,b) such that g'(c) = 0
(x)(f'(x)) - f(x)
g'(x) = -------------------
x^2
(c)(f'(c)) - f(c)
=> g'(c) = ------------------- = 0
c^2
=> (c)(f'(c)) - f(c) = 0
f(c)
=> f'(c) = ------ = k
c
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