Maximum no. of critical points that a polynomial of degree n can have?Why?

2 Answers
Mar 25, 2018

It can have (n-1) critical points

Explanation:

A polynomial function of degree n can have at most n-1 critical points while the least number is 1 depending on the function.

A first-degree polynomial function has no critical points as it's represented by a straight line

A second-degree polynomial function has only 1 critical point.

A third-degree polynomial function can have 1 or 2(which is (n-1)).

I hope this helped.

Mar 25, 2018

#n -1#

Explanation:

Suppose that #f(x) = ax^n + bx^(n - 1) + cx^(n - 2) + ... + n#

Now recall that critical points occur when the derivative equals #0#.

#f'(x) = anx^(n -1) + b(n - 1)x^(n - 2) + ... + n#

If we try to solve the equation

#0 = anx^(n - 1) + b(n - 1)x^(n - 2) + ... + n#

We see that the maximum number of solutions is the degree of the highest degree term of the derivative, which will be #n - 1#.

Hopefully this helps!