Solve Tangent Lines Problems in Calculus
Tangent lines problems and their solutions, using first derivatives, are presented.
 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:
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:
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 
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