Suppose you have a graph of a polynomial function and you can see that the function increases without bound on both left and right ends, has 4 real zereos and has 5 turning points. Based on this information, what is the minimum degree of the polynomial?
So the 1st derivative has 5 solutions, indicating that the 1st derivative must have been a 5th degree, making the original function sixth degree function
It must have had some double roots, that is , touching the x-axis without crossing over gives us 2 equal roots.
To determine the minimum degree of the polynomial, we can use the given information.
We know that the function increases without bound on both the left and right ends. This suggests that the leading term of the polynomial has a positive coefficient, indicating that the degree of the polynomial is even.
Additionally, we are told that the function has 4 real zeros. By the fundamental theorem of algebra, a polynomial of degree n can have at most n real zeros. Since we have 4 real zeros, the degree of the polynomial must be at least 4.
Finally, we are given that the function has 5 turning points. A turning point is a point where the function changes its direction from increasing to decreasing or vice versa. Each turning point corresponds to either a local minimum or a local maximum. For a polynomial of degree n, there can be at most n - 1 turning points. Since we have 5 turning points, the degree of the polynomial must be at least 6.
Therefore, based on the given information, the minimum degree of the polynomial is 6.