Answer to Question #242817 in Calculus for Anand

Question

Which of the following statements are true and which are false? Give reasons for your
answers, in the form of a short proof or a counter example.
(i) The function f defined by f(x) = tan(2x) is a periodic function with period π.

(ii) The function f: R---> R defined by f(x) = 1-|x| is differentiable at x=1

(iii) The function f: [3,4] ----> R defined by
f(x) = x

Answer 1

(i) Function f(x) = tanx is a periodic function with period $\pi$.

So, period of tan2x is $\pi$/2. This is because period of tanax is $\pi$/a.

The given statement is false.

(ii) Lets graph f(x) = 1 - |x|

Clearly, at x = 1, there is only tangent. So, the function is differentiable at x = 1.

The given statement is true.

(iii) Lets graph the function f(x) = x

Clearly, the function is defined in the given domain [3, 4].

So the given statement is correct.

The final graph of the function is given as,