推 andy26883372:謝謝你的熱心幫忙 218.169.54.49 01/12 23:30
※ 引述《andy26883372 (fly)》之銘言：
: 謝謝幫忙。
: (a) If f is a one-to-one, twice differentiable function with inverse
: function g, show that
: f"(g(x))
: g"(x)= _ --------------------------
: [f'(g(x))]^3
: (b) Deduce that if f is increasing and concave upward, then its inverse
: function is concave downward.
(a)
From Larson 8th textbook
Theorem 5.9
The Derivative of an Inverse Function
Let f be a function that is differentiable on the interval I.
If f has a inverse function g, then g is differentiable at any x
for which f'(g(x)) not equal to 0, moreover
1
g'(x) = _____________
f'(g(x))
-1
If y = g(x) = f (x) , then f(y) = x
and f'(y) = dx/dy
1 1 1
g'(x) = dy/dx = ___________ = __________ = ________
f'(g(x)) f'(y) dx/dy
g"(x) = d^2y/dx^2 = d -[f'(y)]'
_____ (dy/dx) = ________________
dx [f'(y)]^2
-f"(y) dy/dx
= _________________
[f'(y)]^2
-f"(y) 1
= __________________ x _____________
[f'(y)]^2 dx/dy
-f"(y) 1
= ___________________ x ___________
[f'(y)]^2 f'(y)
-f"(y)
= ______________________
[f'(y)]^3
-f"(g(x))
= _____________________ Q.E.D.
[f'(g(x))] ^3
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