Question

x | f(x)
-4 | 105
-3 | 0
-2 | -15
-1 | 0
0 | 9
1 | 0
2 | -15
3 | 0
4 | 105
5 | 384
according to the table, which ordered pair is a local minimum of the function, f(x)?
(0, 9)
(4, 105)
(-1, 0)
(2, -15)

Explanation:

Step1: Recall local minimum definition

A local minimum of a function is a point where the function value is less than or equal to the function values at neighboring points. We check each ordered pair by looking at the function values around the x - value in the table.

Step2: Analyze (0, 9)

For \(x = 0\), the function value \(f(0)=9\). The values around \(x = 0\) are \(f(- 1)=0\) and \(f(1) = 0\). Since \(0<9\), \(f(0) = 9\) is not a local minimum.

Step3: Analyze (4, 105)

For \(x = 4\), \(f(4)=105\). The value before \(x = 4\) is \(f(3)=0\) and after \(x = 4\) is \(f(5)=384\). Since \(0 < 105\), \(f(4)=105\) is not a local minimum.

Step4: Analyze (-1, 0)

For \(x=-1\), \(f(-1) = 0\). The values around \(x=-1\) are \(f(-2)=- 15\) and \(f(0)=9\). Since \(f(-2)=-15<0\), \(f(-1) = 0\) is not a local minimum.

Step5: Analyze (2, -15)

For \(x = 2\), \(f(2)=-15\). The values around \(x = 2\) are \(f(1)=0\) and \(f(3)=0\). Since \(-15<0\) (both \(f(1)\) and \(f(3)\) are 0, and \(-15\) is less than 0), \(f(2)=-15\) is a local minimum as it is less than the function values at its neighboring points (\(x = 1\) and \(x = 3\)).

Answer:

(2, -15)