Derivative Problems

1Find the point in the function y = |x + 2| where it has no derivative. Justify the result by representing it graphically.

2Find the point in the function y = |x 2 − 5x + 6| where it has no derivative. Justify the result by representing it graphically.

3Study the continuity and differentiability of the function defined by:

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4Given the function:

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For what values of a is the function differentiable?

5Determine the values of a and b where the following function is continuous and differentiable:

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6Determine the values of a and b for which the function is differentiable at all points:

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7Find the points where y = 250 − |x² −1| has no derivative.

8Determine for which values of a and b the function is continuous and differentiable:

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1

Find the point in the function y = |x + 2| where it has no derivative. Justify the result by representing it graphically.

Function

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The function is continuous.

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f'(−2) = −1f'(−2)+ = 1

It has no derivative at P(−2,0).

Graph of Function


2

Find the point in the function y = |x 2 − 5x + 6| where it has no derivative. Justify the result by representing it graphically.

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The function is continuous.

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f'(2)- = −1f'(2)+ = 1

f'(3)- = −1f'(3)+ = 1

The function is not differentiable at: x = 2 and x = 3 or at points P1(2,0) and P2(3,0).

Graph of Function


3

Study the continuity and differentiability of the function defined by:

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The function is not continuous at x = 0 because it has no image. Therefore it is not differentiable.

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The function is continuous.

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The function is not differentiable at any point.


4

Given the function:

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For what values of a is the function differentiable?

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Differentiable at a = 1

For x = −1, it is not continuous.


5

Determine the values of a and b where the following function is continuous and differentiable:

Function

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Solution


6

Determine the values of a and b for which the function is differentiable at all points:

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A differentiable function has to be continuous. In this case the function is not continuous for x = 0, that is to say, there are no values for a and b which make the function continuous.

Therefore, there are no values of a and b for which the function is differentiable.


7

Find the points where y = 250 − |x² −1| has no derivative.

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The function is continuous.

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Is not differentiable at x = −1 and x = 1.


8

Determine for which values of a and b the function is continuous and differentiable:

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For a = 1 and b = 4, the function is continuous.

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It is not differentiable at x = 0.

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It is differentiable at x = 2.




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