Limits And Derivations

To find the points where the tangents to the curve are parallel to the x-axis, we need to determine where the derivative of the function is zero.

First, let's find the derivative of the given function:

The derivative is:

We set the derivative equal to zero to find the critical points:

Factor out the common term:

This gives us:

or

Solving the quadratic equation :

By factoring, we get:

Thus, or .

So the critical points are and .

Now, find the corresponding values for each :

• For :
  
  Point: (0, -4)
• For :
  
  Point: (1, -3)
• For :
  
  Point: (2, -4)

Therefore, the points where the tangents to the curve are parallel to the x-axis are (0, -4), (1, -3), and (2, -4).

The correct answer is: (0, -4), (2, -4) and (1, -3).

Factorize both the numerator and denominator:

So, the expression becomes:

Cancel out the terms:

Substitute :