Explanation:
To find the equation of the polynomial function, we can use the factored form, which involves multiplying out factors of the form (x - a), where a is a root of the polynomial. Since f(x) has x-intercepts at -3, 0, and 3, we know that the function can be written as:
f(x) = A(x + 3)(x - 3)x
where A is some constant that scales the function vertically. To find A, we can use one of the local extrema. Since f(x) has a local maximum at x = 2, we know that f'(2) = 0 and f''(2) < 0. Taking derivatives of f(x), we get:
f'(x) = A(x + 3)(x - 3) + A(x - 3)x + A(x + 3)x = 3Ax
f''(x) = 6A
Setting f'(2) = 0, we get:
f'(2) = 3A(2) = 0
A = 0
This tells us that the local maximum at x = 2 is actually a horizontal inflection point. To find the value of the function at x = -1 (the local minimum), we can substitute this value into the factored form and simplify:
f(-1) = A(-1 + 3)(-1 - 3)(-1) = 8A
Since we want the function to have a local minimum at x = -1, we also want f''(-1) > 0. Using the second derivative test, we can see that this condition is satisfied if f''(x) > 0 for x < -1 and f''(x) < 0 for x > -1. Since f''(x) is constant, we just need to check its sign at any point:
f''(-2) = 6A > 0
A > 0
Therefore, we can choose A = 1 and write the equation of the polynomial function as:
f(x) = (x + 3)x(x - 3)
or
f(x) = x^3 - 9x