# Derivative Problems

### Solutions

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:

4Given the function:

For what values of **a** is the function differentiable?

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

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

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:

## 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) ^{−} = −1**

**f'(−2)**

^{+}= 1It has no derivative at P(−2,0).

## 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) ^{-} = −1**

**f'(2)**

^{+}= 1**f'(3) ^{-} = −1**

**f'(3)**

^{+}= 1The function is not differentiable at: x = 2 and x = 3 or at points P_{1}(2,0) and P_{2}(3,0).

## 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.

## 4

Given the function:

For what values of **a** is the function differentiable?

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:

## 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.

## 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.

## 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.