※ 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 -- Ask not what Taistock can do for you! Ask what you can do for Taistock! --

※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.112.7.59