Alegra 2 problem? Turning points in a graph.

Question

What is the max number of turning points in the graph of P(x) = 7x5 + 14x4 – 7x³?
Question options:

a) 1

b) 2

c) 4

d) 5

Answer 1

 

Hi Terry,

 

If the first derivative of a function changes sign at the point where it is zero,

 

that point is called a turning point, and

 

that point can be either a local maximum or a local minimum.

 

In this problem,

 

     P'(x) = 35x⁴ + 56x³ - 21x²

 

             = 7x²(5x² + 8x - 3)

 

So, P'(x) = 0 at x = (-4-√31)/5, 0, (-4+√31)/5.

 

P'(x) changes its sign

 

at x = (-4-√31)/5 from positive to negative and

 

at x = (-4+√31)/5 from negative to positive.

 

So, P(x) has 2 turning points.

 

 

 

 

 

Answer 2

For a polynomial of degree n, the maximum number of turning points is (n - 1).

 

This polynomial has degree 5, so the answer is 4.