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Calculus Level 5

If $f(x)$ and $g(x)$ are functions defined in the real domain and co-domain, such that $\sqrt{1-(f(x))^2}=g(x)$ , which of the following statements are necessarily true?

A: If $g(x)$ is periodic with period 1, $f(x)$ is periodic with period $\dfrac{1}{2}$
B: If $g(x)$ is a continuous function, $f(x)$ is also continuous in their respective domains.
C: If $-f(c)=f'(c)=0.5;\dfrac{g'(c)}{g(c)}=\dfrac{1}{3}$ .
D: If $g(x)$ is an even function, $f(x)$ is odd.
E: If $g(x)$ is an odd function, $f(1024)$ is of unit modulus.

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Sarthak Agrawal
Mar 20, 2018

A : Counterexample $sin(2\pi x)$ and $\mid cos(2\pi x) \mid$

B : Statement is correct for $\mid f(x)\mid$ but not correct currently. Suppose at any instant $f(x) = 0.5$ and at the next instant $f(x) = -0.5$ , though $g(x)$ would still remain continuous $f(x)$ is not continuous.

C :Take logarithm both sides and differentiate once to obtain directly the required expression.

D :Nonsense.

E :Yes because g(x) = 0 is the only odd fuction possible for g(x). Think why!

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