#### To determine

**To check:** Whether the function f(x) is differentiable at the given point or not and then sketch the graphs of f(x) and f′(x).

#### Answer

The function f(x) is not differentiable at 1.

#### Explanation

**Given:**

The function is f(x)={x2+1 if x<1x+1 if x≥1.

The point is x=1.

**Calculation:**

Obtain the left hand derivative of f(x) at 1.

That is, Compute limh→0−f(1+h)−f(1)h .

Since f(x)=x2+1, f(1+h)=(1+h)2+1 and f(1)=12+1.

limh→0−f(1+h)−f(1)h=limh→0−[(1+h)2+1]−[12+1]h=limh→0−[1+h2+2h+1]−(1+1)h=limh→0−h2+2h+2−2h=limh→0−h2+2hh

Divide the numerator and the denominator by *h*.

limh→0−f(1+h)−f(1)h=limh→0−h+21=limh→0−h+2=2

Thus, the value of the left hand derivative of f(x) at 1 is 2.

Obtain the right hand derivative of f(x) at 1.

That is, compute limh→0+f(1+h)−f(1)h.

Since f(x)=x+1, f(1+h)=(1+h)+1 and f(1)=1+1.

limh→0+f(1+h)−f(1)h=limh→0+(h+2)−2h=limh→0+hh=limh→0+1=1

Thus, the value of the right hand derivative of f(x) at 1 is 1.

Since the value of the left hand derivative of f(x) and the value of the right hand derivative of f(x) are not equal, the limit f′(1)=limh→0f(1+h)−f(1)h does not exist.

Therefore, the function f(x) is not differentiable at 1.

It is required to sketch the function f(x) and f′(x).

**Graph:**

The graph of the function f(x) is shown below in Figure 1.

From Figure 1, it is observed that the function is a parabola when x<1 and the function is a straight line when x≥1.

The graph of the function f′(x) is shown below in Figure 2.

From Figure 2, it is observed that the derivative of f(x) is defined only x<1 and x>1. But, the derivative of f(x) is not defined at x=1.