The third degree polynomial function is , .

.

Apply derivative on each side with respect to .

.

Slope of the tangent line is derivative of the function.

(a)

The function has no horizontal tangent.

Horizontal tangents will occur where the derivative of the function is zero.

Since the function has no horizontal tangents, .

The function has no horizontal tangent, when there are no real solutions for .

Find the roots of the polynomial using formula .

has no real solutions if and only if and .

for .

If , then .

The function has no horizontal tangent for and for .

(b)

The function has one horizontal tangent.

The function has one horizontal tangent, has for one of the values of .

The function has one horizontal tangent, when there are two real solutions but not identical for .

has two real solutions when .

The function has one horizontal tangent when .

(c)

The function has two horizontal tangent.

The function has two horizontal tangent, has two real and unique solution of .

has two real and unique solutions when .

for .

for .

The function has two horizontal tangent for and for .

(a) The function has no horizontal tangent for and for .

(b) The function has one horizontal tangent when .

(c) The function has two horizontal tangent for and for .