The limacon curve is .

Consider .

has at most two solutions in its period.

Range of sine function is .

and .

Thus, expression implies that .

The polar equation is .

For an inner loop .

.

The inner loop begins at and ends at .

For an inner loop .

.

The inner loop begins at and ends at .

Insert .

.

Substitute , in .

Set slope to zero to find what values of results zero slopes.

or .     

and

and

The two solutions for inner loop, corresponding to the horizontal slope are

and .

(b)

For example consider the value of greater than or less than .

Prove that there is a dimple means that there is no horiontal slope for greater than .