**EXERCISE 4.2**

3. (a)

**First derivative test**

Differentiating (1) w.r.t .

Given that

Sign analysis of

By first derivative test, has relative minimum at

**Second derivative test**

differentiating (2) w.r.t .

Thus by second derivative test has relative minimum at .

3. (b)

**First derivative test**

Differentiating (1) w.r.t

Given that

Sign analysis of

By the first derivative test we conclude that has relative minimum at

**Second derivative test**

Differentiating (2) w.r.t

Giver what

has relative maximum at .

4(a)

Given that

**First derivative test**

Sign analysis of ( For test value close to o).

By fist derivative we conclude that has relative minimum at .

**second derivative test**

Differentiating (1) w.r.t.

Given what

has relative minimum at

4.(b)

Given

**First derivative test**

Sign analysin of near to 0 .

By the first derivative test we come to the conclusion that has relative minimum at

**Second derivative test**

Differentiating (1) w.r.t.

has relative minimum at

(C) Both the functions and are squares for and near to 0 . Both have positive values for values of close to zero and both are zero at .

5. (a)

will be stationary point of and if and

is a stationary point of and .

(b) Differentiating (1) and (2) in (a)