Question

Which of the following statements are true? Give reasons for your answers, in the form of a short proof or a counter example.
(1) d^2/dx^2=(dy/dx) ^2
(2) The inverse function of y=e^3x is y=1/3lnx.
(3) If f is increasing and f(x) >0 on an interval I, then g(x) = 1/f(x) is deceasing on I.
(4) An equation of the tangent line to the parabola y=x^2 at (-2, 4) is you=2x(x+2).
(5) If f is one - one onto and differentiable on R, then (f^-1) '(6) =1/f'(6) .

Answer 1

(1) It's false. For example, if $y=x$ then $\frac{d^2y}{dx^2}=0$ and $(\frac{dy}{dx})^2=1$

(2) It's false. $\ln y=3x$ hence $x=\frac{\ln y}{3}$ then the inverse function $y=\frac{\ln x}{3}$

(3) It's true. If $f$ is increasing hence $f'>0$ then $g'(x)=(\frac{1}{f(x)})'=-\frac{f'(x)}{(f(x))^2}<0$ hence $g(x)$ is decresing on I

(4) It's false. The tangent line is $y-4=y'(x+2)$ as $y'=2x$ hence $y=2x(x+2)+4$

(5) It's false. For example $f=x$ then $f^{-1}=\frac{1}{x}$ then $f'(6)=1$ and $(f^{-1})'(6)=-\frac{1}{36}$