(a)

The functions are and .

indicates that the critical point is at .

Here the function has either local maximum or local minimum.

From the second derivative test, curve is concave down.

Therefore local maximum at .

(b)

The functions are and .

which indicates that the critical point is at .

Here the function has either local maximum or local minimum.

From the second derivative test,

, curve is neither concave up nor concave down.

Therefore conclusion cannot be made at .

The function has a horizontal tangent because slope is zero.

(a) The function has a critical point at and it has a local maximum.

(b) The function has a horizontal tangent at .