# Thread: Differentiability: f(x) = cx+d for x<=2, x^2-cx for x>2

1. ## Differentiability: f(x) = cx+d for x<=2, x^2-cx for x>2

If
$f(x)= \left\{ \begin{array}{c}cx+d, \mbox{ } x\leq 2\\ x^2-cx, \mbox{ } x>2$
and f'(x) is defined at x=2, what is the value of c+d?

2. ## Re: Differentiability

a function is continuous at x = a if ...

1. f(a) is defined.
2. lim{x->a} f(x) exists.
3. lim{x->a} f(x) = f(a)

use this definition of continuity for the functions f(x) and f'(x) ... you'll find c and d.

3. ## Re: Differentiability

I tried that but then I got the equation
$2c+d=4-2c$
which is where I got stuck.

4. ## Re: Differentiability

The derivative should also be equal at x = 2:
$f&#39;(x) = \left\{ \begin{array}{ll} c & x \leq 2 \\ 2x - c & x > 2$

Since f'(x) is defined, the deriative at x = 2 can't simultaneously hold 2 values so you can solve for c and then plug it back into the original equation to solve for d.

5. ## Re: Differentiability

Originally Posted by tjkubo
I tried that but then I got the equation
$2c+d=4-2c$
which is where I got stuck.
you did it for f(x) ... did you do it for f'(x)?

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