Question

F(x) = (x³) * (x - 1)⁵ Find the zero(s) at which flattens out express the zero(s) as ordered pair.

Options:
a) (0,0), (1,0)
b) (0,0)
c) (1,0)
d) No zeros

Answer 1

To find the zero(s) at which the function F(x) = (x³) * (x - 1)⁵ flattens out, we differentiate and set the derivative to zero. The zero(s) where the derivative equals zero are x = 0 and x = 1, leading to the ordered pairs (0,0) and (1,0), corresponding to option (a).

Step-by-step explanation:

The question asks to find the zero(s) at which the function F(x) = (x³) * (x - 1)⁵ flattens out and to express the zero(s) as ordered pairs. A function flattens out, or has a horizontal tangent, where its derivative is equal to zero. So, we need to find the values of x for which the derivative of F(x) is zero.

To find the derivative, apply the product rule:

1. 
   
3. 
   

So, the derivative of F(x) is F'(x) = 3x²(x - 1)⁵ + x³ * 5(x - 1)⁴. Setting F'(x) = 0 gives us the zeroes of the derivative, which are x = 0 and x = 1. These are the points where the function flattens out.

Therefore, the function flattens out at the ordered pairs (0,0) and (1,0), which correspond to option (a).