Question

values of the polynomial function f for selected values of x are given in the table. if all of the zeros of the function f are given in the table, which of the following must be true? a the function f has a local minimum at (-1, -36). b the function f has a local minimum at (5, 0). c the function f has a local maximum at (3, 4). d the function f has a local maximum at (1, 0).

Explanation:

Step1: Recall local - minimum and maximum definition

A local minimum occurs when the function value at a point is less than or equal to the function values at nearby points. A local maximum occurs when the function value at a point is greater than or equal to the function values at nearby points.

Step2: Analyze the values around each point

For \(x=-1\), we only know the value at \(x = 1\) to the right. Since \(f(1)=0>-36 = f(-1)\), we can't be sure it's a local minimum.
For \(x = 1\), \(f(-1)=-36<0=f(1)\) and \(f(3)=4 > 0\), so it's not a local maximum.
For \(x = 3\), \(f(1)=0<4=f(3)\) and \(f(5)=0<4\), but we don't know the values in a small - enough neighborhood around \(x = 3\) to be sure it's a local maximum.
For \(x = 5\), \(f(3)=4>0=f(5)\) and \(f(7)=12>0=f(5)\). Since the function value at \(x = 5\) is less than the function values at nearby points (the points we know), the function \(f\) has a local minimum at \((5,0)\).

Answer:

B. The function \(f\) has a local minimum at \((5,0)\)