 Problem 1: Find all points on the graph of y = x^{ 3}  3 x where the tangent line is parallel to the x axis (or horizontal tangent line).
Solution to Problem 1:

Lines that are parallel to the x axis have slope = 0. The slope of a tangent line to the graph of y = x^{ 3}  3 x is given by the first derivative y '.
y ' = 3 x^{ 2}  3

We now find all values of x for which y ' = 0.
3 x^{ 2}  3 = 0

Solve the above equation for x to obtain the solutions.
x =  1 and x = 1

The above values of x are the x coordinates of the points where the tangent lines are parallel to the x axis. Find the y coordinates of these points using y = x^{ 3}  3 x
for x =  1 , y = 2
for x = 1 , y =  2

The points at which the tangent lines are parallel to the x axis are: (1 , 2) and (1 , 2). See the graph of y = x^{3}  3 x below with the tangent lines.
Problem 2: Find a and b so that the line y =  3 x + 4 is tangent to the graph of y = a x^{3} + b x at x = 1.
Solution to Problem 2:

We need to determine two algebraic equations in order to find a and b. Since the point of tangency is on the graph of y = a x^{3} + b x and y =  3 x + 4, at x = 1 we have
a(1)^{3} + b(1) =  3(1) + 4

Simplify to write an equation in a and b
a + b = 1

The slope of the tangent line is 3 which is also equal to the first derivative y ' of y = a x^{3} + b x at x = 1
y ' = 3 a x^{2} + x =  3 at x = 1.

The above gives a second equation in a and b
3 a + b = 3

Solve the system of equations a + b = 1 and 3 a + b =  3 to find a and b
a =  2 and b = 3.

See graphs of y = a x^{3} + b x, with a =  2 and b = 3, and y =  3 x + 4 below.
Problem 3: Find conditions on a and b so that the graph of y = a e^{ x} + bx has NO tangent line parallel to the x axis (horizontal tangent).
Solution to Problem 3:

The slope of a tangent line is given by the first derivative y ' of y = a e^{ x} + bx. Find y '
y ' = a e^{ x} + b

To find the x coordinate of a point at which the tangent line to the graph of y is horizontal, solve y ' = 0 for x (slope of a horizontal line = 0)
a e^{ x} + b = 0

Rewrite the above equation as follows
e^{ x} =  b/a

The above equation has solutions for a/b >0. Hence, the graph of y = a e^{ x} + bx has NO horizontal tangent line if a/b <= 0
Exercises
1) Find all points on the graph of y = x^{ 3}  3 x where the tangent line is parallel to the line whose equation is given by y = 9 x + 4.
2) Find a and b so that the line y =  2 is tangent to the graph of y = a x^{2} + b x at x = 1.
3) Find conditions on a, b and c so that the graph of y = a x^{ 3} + b x^{ 2} + c x has ONE tangent line parallel to the x axis (horizontal tangent).
solutions to the above exercises
1) (2 , 2) and (2 , 2)
2) a = 2 and b =  4
3) 4 b^{ 2}  12 a c = 0
More references on
calculus problems 