(a)

Thu function is , .

Graph :

Graph the function  :

Observe the garph.

The function has local maximum at  is .

The function has local maximum at  is .

Local maximum is .

Local minimum is .

Thu function is .

Differentiate  on each side with respect to .

Find the critical points.

Thus, the critical points exist when .

Equate  to zero.

The general solution for sine function is , where .

The solution for  is , .

The critical point is at  and .

Substitute  in .

Local maximum is .

Substitute  in .

Local minimum is .

(b)

Thu function is .

Obsderve the graph.

The function  most rapidly increases over the interval .

Now find . 

. 

Differentiate  on each side with respect to .

.

Find the exact value where  increases.

Now equate  to zero and plug the value into .

The general solution of cosine function is , where 

The solution for  is , .

 and .

Substitute  in .

Substitute  in .

Therefore the function is increases at .

(a)

Local maximum is .

Local minimum is .

(b)

The function is increases at .