Derivative Problems

Solution of exercise 1

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

The function is continuous.

f'(−2)⁻ = −1f'(−2)⁺ = 1

It has no derivative at P(−2,0).

Solution of exercise 2

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

The function is continuous.

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).

Solution of exercise 3

Study the continuity and differentiability of the function defined by:

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

The function is continuous.

The function is not differentiable at any point.

Solution of exercise 4

Given the function:

For what values of a is the function differentiable?

Differentiable at a = 1

For x = −1, it is not continuous.

Solution of exercise 5

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

Solution of exercise 6

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

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.

Solution of exercise 7

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

The function is continuous.

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

Solution of exercise 8

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

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

It is not differentiable at x = 0.

It is differentiable at x = 2.