Use the following partial graph of a polynomial function, with a local maximum at x=-4 and a local minimum at x=2

Question

Compare f(1.9) and f(2.1) with f(2)

Answer 1

For f(x) to have a local minimum at x = 2, it must be of the form g(x)*(x - 2)^2 + f(2), where g(x) is itself a polynomial function of x, and g(2) > 0. Without seeing the graph of f, or knowing what f(2) is, one can't say much more, but, because 2 is a local minimum, f(1.9) and f(2.1) must be ever so slightly greater than f(2) (unless f has other local extrema between 1.9 and 2.1), perhaps only a few hundredths greater.

Since f also has a local maximum at x = - 4, it must also be of the form h(x)*(x + 4)^2 + f(- 4), where h(x) is another polynomial in x and h(- 4) < 0.Unfortunately, I cannot say anything more about f because I am not given f(2), f(- 4), or the degree of f.