Maximum no. of critical points that a polynomial of degree n can have?Why?
2 Answers
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.
Explanation:
Suppose that
Now recall that critical points occur when the derivative equals
#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
Hopefully this helps!