#### To determine

**To Show:** The function g′(0) does not exist.

#### Explanation

**Result Used:** The derivative of a function at x=a is given by formula**,**

g′(a)=limx→ag(x)−g(a)x−a (1)

**Proof:**

Consider the function, g(x)=x23.

Compute g′(0) by using the equation (1),

g′(0)=limx→0g(x)−g(0)x−0=limx→0x23−(0)23x=limx→0x23x=limx→0x⋅x−13x

=limx→01x13

Here, the function 1x13 becomes larger and larger as *x* tends to zero. That is,

g′(0)=limx→01x13=∞

Therefore, the derivative of the function does not exist at x=0.

Thus, the required proof is obtained.

#### To determine

**To find:** The value of g′(a) if a≠0.

#### Answer

The value of derivative of f(x) at x=a is 23a13.

#### Explanation

**Given:**

The function y=x23.

**Calculation:**

Obtain the derivative of the function g(x) at x=a.

Compute g′(a) by using the equation (1),

g′(x)=limx→ag(x)−g(a)x−a=limx→ax23−a23x−a=limx→a(x13)2−(a13)2(x13)3−(a13)3=limx→a(x13+a13)(x13−a13)(x13−a13)(x23+a23+x13a13)

Simplify further,

g′(a)=limx→a(x13+a13)(x23+a23+x13a13)=((a)13+a13)((a)23+a23+(a)13a13)=2a133a23=23a13

Thus the value of the derivative at x=a is, g′(a)=23a13.

#### To determine

**To Show:** The y=x23 has a vertical tangent line at (0,0).

#### Explanation

**Result Used:**

A curve has a vertical tangent line at x=a if *g* is continuous at x=a and limx→a|g′(x)|=∞

**Proof:**

Consider the equation y=x23.

Substitute x=0 in g(x),

g(0)=(0)23=0

Thus g(0)=0 is defined.

The limit of the function g(x) is computed as follows,

limx→0g(x)=limx→0x23=(0)23=0

Therefore, g(x) is continuous at x=0.

From part (b), g′(x)=2x−133.

Take the limit of the function g′(x) as *x* approaches zero.

limx→0|g′(x)|=limx→0|2x−133|=23limx→0|x−13|=23limx→0|1x13|=23limx→01|x13|

=∞

Since the function g(x) is continuous at x=0 and limx→0|g′(x)|=∞.

By result, the curve g(x) has a vertical tangent at x=0.

Thus, the curve g(x) has a vertical tangent at the point (0,0).

#### To determine

**To illustrate:** Thepart (c) by graphing y=x23.

#### Explanation

**Graph:**

Use the online graphing calculator to draw the graph of the function y=x23 as shown below in Figure 1,

**Illustration:**

From Figure 1, it is clear that the *y*-axis is the vertical tangent to the curve y=x3.