Question

x | f(x)
-2 | 0
-1 | 45
0 | 64
1 | 45
2 | 0
3 | -35
4 | 0
5 | 189
6 | 640
according to the table, which ordered pair is a local maximum of the function, f(x)?
(0, 64)
(3, -35)
(5, 189)
(2, 0)

Explanation:

Step1: Recall the definition of local maximum

A local maximum of a function \(y = f(x)\) is a point \((a,f(a))\) such that \(f(a)\geq f(x)\) for all \(x\) in some open interval containing \(a\).

Step2: Analyze the values around each \(x\) - value

• For \(x = 0\):

We look at the values of \(f(x)\) for \(x=-1\) and \(x = 1\). \(f(-1)=45\), \(f(0) = 64\), \(f(1)=45\). Since \(45<64\), \(f(0)\geq f(x)\) for \(x=-1\) and \(x = 1\) (in the interval \((-1,1)\)).

• For \(x = 3\):

\(f(2)=0\), \(f(3)=- 35\), \(f(4)=0\). Since \(-35<0\), it is not a local maximum.

• For \(x = 5\):

We need to check the values around \(x = 5\). But we only know \(f(4) = 0\) and \(f(6)=640\). Since \(0<189<640\), it is not a local maximum (because \(f(6)>f(5)\)).

• For \(x = 2\):

\(f(1)=45\), \(f(2)=0\), \(f(3)=-35\). Since \(0<45\), it is not a local maximum.

Answer:

\((0,64)\)