1i think the options u provided r themselves incorrect
it should be 6Ï€2 +2 or 2(3Ï€2 +1) in option A which is the correct answer
f(x) = (2x-Ï€)3 + 2x - cos x
the value of ddx(f-1x) at x=Ï€
(a) 3Ï€2 + 2
(b) -2
(c) 13Ï€2 + 2
(d) 13
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3 Answers
1
23f^{-1}(x)=g(x), \Rightarrow f(g(x))=x
sof^{!}(g(x))g^{!}(x)=1
g^{!}(x)=\frac{1}{f^{!}(g(x))}
at , g^{!}(\pi )=\frac{1}{f^{!}(g(\pi ))}
now f(g(\pi ))=\pi
notice that f(\frac{\pi }{2})=\pi
hence
g(\pi )=\frac{\pi }{2}
( since f has to be one one for its inverse to be defined )
hence g^{!}(\pi )=\frac{1}{f^{!}(\frac{\pi }{2})}
{f^{!}(\frac{\pi }{2})}=3, hence
g^{!}(\pi )=\frac{d}{dx}(f^{-1}(x))=\frac{1}{f^{!}(\pi /2)}=\frac{1}{3}
