- #1

opus

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## Homework Statement

Sketch the graph of the following function and use the definition of the derivative to show that the function is not differentiable at x=1.

$$f(x) = \begin{cases} {-x^2+2} & \text{if } x \leq 1 \\ x & \text{if } x > 1 \end{cases}$$

## Homework Equations

Derivative: $$f'(x) = \lim_{h \rightarrow 0} {\frac{f(x+h) - f(x)}{h}}$$

## The Attempt at a Solution

Now I graphed this peicewise function and there is a sharp kink at x=1.

Visually, I know that there is no derivative here, because if we think about tangent line to the graph at the point of the kink, it can swivel around and there is no definite derivative.

Algebraically, I'm not sure what to say.

The point is continuous, but a point can be continuous and not be differentiable, but not the other way around.

I took the derivative function at ##x\leq 1## and that derivative function is ##f'(x)= -2x##

I took the derivative function at ##x>1## and that derivative function is ##f'(x) = 1##

So this tells me that the derivative functions are different at the left of x=1 and at the right, which is obvious because the piecewise has two separate function on these intervals anyways.

So I'm at a standstill on what I need to say mathematically to show that the function is not differentiable at x=1.