The mass of the part of the metal rod is .

Linear density is the derivative of mass with respect to position .

.

Substitute  in above expression.

The linear density is .

(a)

When  m :

The linear density is .

Substitute  in above expression.

 kg/m.

(b)

When  m :

The linear density is .

Substitute  in above expression.

 kg/m.

(c)

When  m :

The linear density is .

Substitute  in above expression.

 kg/m.

Therefore, at the right end density is highest and at the left end density is lowest.

(a)  kg/m.

(b)  kg/m.

(c)  kg/m.

At the right end density is highest and at the left end density is lowest.