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.