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 April 23rd, 2016, 08:43 AM #1 Newbie   Joined: Apr 2016 From: Sweden Posts: 2 Thanks: 0 When is this sine function differentiable at all points? The question is: For which values of a and b is the following function differentiable at all points? $$f(x)=\sin(|x^2+ax+b|)$$ I'm new to these kind of problems, so any help is deeply appreciated. Thanks in advance.
 April 23rd, 2016, 09:02 AM #2 Senior Member     Joined: Dec 2013 From: some subspace Posts: 212 Thanks: 72 Math Focus: real analysis, vector analysis, numerical analysis, discrete mathematics For function to be differentiable, what it requires from the argument of abs-function? For example plot $\displaystyle |x|$ and $\displaystyle |-1 - x^4|$ to get some idea. Then, use that knowledge to determine what this means for the parabola in question.
April 23rd, 2016, 09:54 AM   #3
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 Originally Posted by fysmat For function to be differentiable, what it requires from the argument of abs-function? For example plot $\displaystyle |x|$ and $\displaystyle |-1 - x^4|$ to get some idea. Then, use that knowledge to determine what this means for the parabola in question.
I understand that the quadratic function isn't differentiable in all points when it is under the x-axis, however I don't know how I should prove that.

 April 23rd, 2016, 10:29 AM #4 Senior Member     Joined: Dec 2013 From: some subspace Posts: 212 Thanks: 72 Math Focus: real analysis, vector analysis, numerical analysis, discrete mathematics If you wish to prove that a function is differentiable, then there is a simple approach: 1. Prove that the function is continuous. 2. Prove that the derivative of the function is continuous i.e. $\displaystyle \lim_{x \to x_0-} f'(x) = \lim_{x \to x_0+} f'(x).$ In your case we know that the function $\displaystyle f(x) = |x^2 + ax + b|$ is continuous in all $\displaystyle x \in \mathbb{R}$. But what happens in the second part? What can you say about abs function when its argument changes sign? And what this implies to coefficients $\displaystyle a$ and $\displaystyle b$? Thanks from greg1313
 April 23rd, 2016, 10:20 PM #5 Math Team   Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 I don't even think proving continuity is required, since differentiability implies continuity.

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