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April 23rd, 2016, 07:43 AM   #1
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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.
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April 23rd, 2016, 08:02 AM   #2
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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.
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April 23rd, 2016, 08:54 AM   #3
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Quote:
Originally Posted by fysmat View Post
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.
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April 23rd, 2016, 09:29 AM   #4
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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 $\displaystyle b$?
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April 23rd, 2016, 09:20 PM   #5
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I don't even think proving continuity is required, since differentiability implies continuity.
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