Suppose that $f\colon \mathbb R\to \mathbb R$ is differentiable at $c$ and that $f(c) := 0 $. Show that $g(x) := |f(x)|$ is differentiable at $c$ if and only if $f'(c) = 0$. My attempt: We know that $f$ is differentiable at $c$ but we do not know what the value of this derivative is so $$\left|...

Prove that the derivative of an even differentiable function is odd, and the derivative of an odd differentiable function is even. Here are my workings so far. Lets prove the derivative of an odd differentiable function is even first. Let the odd function be $f(x)$. We have $f(-x)=-f(x)$ and...

If $f : \mathbb{R} \to \mathbb{R}$ is differentiable at $c \in \mathbb{R}$, show that $f'(c) =\lim (n(f(c + 1/n) - f (c))$ However, show by example that the existence of the limit of this sequence does not imply the existence of $f'(c)$ I want to know the only second part of the question. I t...

Given a $4\times4$ real matrix $A$, let $T:\mathbb R^4\to \mathbb R^4$ be the linear transformation defined by $Tv=Av$, where we think of $\mathbb R^4$ as the set of real $4\times1$ matrices. For which choices of $A$ given below, do the $Image(T)$ and $Image(T^2)$ have respective dimension 2 a...