True or False? #1

Calculus Level 4

Which of the followings are true?

A. Let $f(x)=x^2+1$ , then $\sqrt{f(x)-f'(x)}$ is differentiable in $(-\infty,~\infty).$

B. For all positive numbers $a$ , function $f(x)=x^a$ that is defined over all reals is differentiable in $(-\infty,~\infty).$

C. If polynomials $f(x)$ and $g(x)$ satisfy $f(g(x))=g(f(x))$ and $f\neq g,$ at least one of $f$ and $g$ is an identity function.

This problem is a part of <True or False?> series .

None of them Only A Only B Only B and C Only C and A Only A and B All of them Only C

1 solution

Boi (보이)
Jul 18, 2017

$f'(x)=2x$ and $\sqrt{f(x)-f'(x)}=\sqrt{x^2-2x+1}=|x-1|$ .

Therefore the expression is not differentiable at $x=1$ .

$\therefore~\boxed{\text{FALSE}}$

(counterexample)

$f(x)=x^{\frac{1}{3}}$ is not differentiable at $x=0$ , because $f'(x)=\dfrac{1}{3x^{\frac{2}{3}}}$ .

$\therefore~\boxed{\text{FALSE}}$

(counterexample)

If $f(x)=2x$ and $g(x)=3x$ , $f(g(x))=6x=g(f(x))$ , but neither $f$ nor $g$ is an identity function.

$\therefore~\boxed{\text{FALSE}}$

Therefore, none of them are true.