isi-entrance 2012 Q22

isi-entrance · India · solved Differentiating Transcendental Functions Regularity and smoothness of transcendental functions

Let $f(x) = x|x|^n$ for $n \geq 1$ a positive integer. Which of the following is true?
(A) $f$ is differentiable everywhere except at $x = 0$
(B) $f$ is continuous but not differentiable at $x = 0$
(C) $f$ is differentiable everywhere
(D) None of the above

$f(x) = x^{n+1}$ for $x > 0$, $f(x) = -x^{n+1}$ for $x < 0$, $f(0) = 0$. Both one-sided derivatives at $0$ equal $0$, so $f$ is differentiable everywhere. (C)

Let $f(x) = x|x|^n$ for $n \geq 1$ a positive integer. Which of the following is true?

(A) $f$ is differentiable everywhere except at $x = 0$

(B) $f$ is continuous but not differentiable at $x = 0$

(C) $f$ is differentiable everywhere

(D) None of the above

Paper Questions

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