Science:Math Exam Resources/Courses/MATH110/December 2011/Question 06
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Find constants a and b such that ƒ is differentiable everywhere. Justify your answer.
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For a function to be differentiable at a point x = a, it must be continuous at a and the derivative on both sides of a must be equal.
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In order for to be differentiable at a, we have two requirements: that the function be continous and derivatives from the right and left of a point are the same.
Before we start, let us note that ƒ is surely, as a composition of differentiable functions, differentiable at all points expect x = a. Hence we can focus our study on the point x = a.
We begin with the second condition: that the derivative from the right () must be the same as the derivative from the left (). This means that we are trying to find the value a such that
Simplifying a bit, this is the same as
If we think about the function we know that it is bounded between -1 and 1. If a was greater than 1/4, say, 1/2, the left side would be less than one, while the right side would already be greater than 2. So a must be between -1/4 and 1/4 for this equation to hold. In fact, the only place that the graphs of and 4x intersect is at x = 0. So a = 0.
This satisfies the condition that the derivatives match; now we must ensure that is continuous. The function is defined everywhere, so we must check that the limit at x = a exists.
The limit from the left of x = 0 is
The limit from the right of x = 0 is
These two limits must be equal, so we have b = 1/2.