Choose k so that f(x)=(2x^2-18)/(x+3), x not 3;=k, x=3 is continuous/differentiable

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PM123

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Is it possible to choose k so that the following function is continuous and differentiable at x = -3?
f(x) = (2x^2 - 18)/(x+3) if x does not equal to -3
k if x = -3

For continuous I'm thinking k = -12 because f(-3) must equal k and the limit as x approaches -3 is -12.

I'm having trouble with differentiable though. The derivative of f(x) is 2, and if k = 2, then the derivative of that is zero, so is it not differentiable?
Thanks.
 
PM123 said:
Is it possible to choose k so that the following function is continuous and differentiable at x = -3?
f(x) = (2x^2 - 18)/(x+3) if x does not equal to -3
k if x = -3

For continuous I'm thinking k = -12 because f(-3) must equal k and the limit as x approaches -3 is -12.

I'm having trouble with differentiable though. The derivative of f(x) is 2, and if k = 2, then the derivative of that is zero, so is it not differentiable?
Thanks.
Click to expand...
How did that come about?

What is the expression for f'(x)?

Does f'(x) depend on 'k'?

If you plot the function from say x = -5 to x =0 - what does it look like?
 
f(x) = (2x^2 - 18)/(x+3)
f(x) = 2(x^2-9)/(x+3)
f(x) = 2(x+3)(x-3)/(x+3)
f(x) = 2(x-3) = 2x-6
f'(x) = 2

It just asks if you can choose k so that the function is differentiable at x = -3.
Graphing the function is just a linear function.
 
PM123 said:
f(x) = (2x^2 - 18)/(x+3)
f(x) = 2(x^2-9)/(x+3)
f(x) = 2(x+3)(x-3)/(x+3)
f(x) = 2(x-3) = 2x-6
f'(x) = 2

It just asks if you can choose k so that the function is differentiable at x = -3.
Graphing the function is just a linear function.
Click to expand...
\(\displaystyle f(x)=\begin{cases}\dfrac{2x^2-18}{x+3} & x\ne-3 \\ -12 & x=-3\end{cases}\)
 
PM123 said:
Yes, that's where it's continuous but is it also where it's differentiable?
Click to expand...
Can you calculate \(\displaystyle \mathop {\lim }\limits_{x \to - 3} f'(x)~?\)

Recall that in \(\displaystyle \mathop {\lim }\limits_{x \to c} g(x)\) it is always the case that \(\displaystyle x\ne c\).
 
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