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April 23rd, 2016, 07:43 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, 08:02 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 absfunction? 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, 08:54 AM  #3 
Newbie Joined: Apr 2016 From: Sweden Posts: 2 Thanks: 0  I understand that the quadratic function isn't differentiable in all points when it is under the xaxis, however I don't know how I should prove that.

April 23rd, 2016, 09:29 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 $\displaystyle b$? 
April 23rd, 2016, 09:20 PM  #5 
Math Team Joined: Nov 2014 From: Australia Posts: 688 Thanks: 243 
I don't even think proving continuity is required, since differentiability implies continuity.


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