Question

use the graph of $y = f(x)$ given below to answer the questions.

list the local maxima as ordered pairs, if any exist. (make sure to enter these as points separated with commas, if needed. write \dne\ if there are no local maxima.)

list the local minima as ordered pairs, if any exist. (make sure to enter these as points separated with commas, if needed. write \dne\ if there are no local minima.)

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Explanation:

Step1: Recall the definition of local maxima

A local maximum is a point where the function changes from increasing to decreasing. Looking at the graph, at \(x = - 2\), the function value is \(y = 6\) (since the point is \((-2,6)\)) and at \(x=2\), the function value is \(y = 0\) (the point is \((2,0)\)). But wait, no, actually, for local maxima, we check the peaks. The peak at \(x=-2\) (point \((-2,6)\)) and also the point \((-1,3)\) is not a peak. Wait, no, re - check:
The local maxima occur where the function has a "peak" in a small interval. The point \((-2,6)\) is a local maximum (since the function is increasing before \(x = - 2\) and decreasing after \(x=-2\)) and the point \((2,0)\) is not a local maximum. Wait, no, another check:
The local maxima: A point \((a,f(a))\) is a local maximum if \(f(a)\geq f(x)\) for all \(x\) in some open interval containing \(a\). The point \((-2,6)\) satisfies this. Also, if we consider the point \((2,0)\), no, because around \(x = 2\), if we take a small interval, say \((1.5,2.5)\), \(f(2)=0\) and \(f(1)= - 1\), \(f(3)=-1.5\) (approximate from the graph). Wait, no, wrong. Wait, the local maxima:
Looking at the graph, the local maxima are the points where the function changes from increasing to decreasing. The point \((-2,6)\) (since to the left of \(x=-2\) (in a small interval), the function is increasing and to the right (in a small interval) it is decreasing) and the point \((2,0)\) is not. Wait, no, another approach:
By the definition of local maxima (a point \((x,y)\) where \(y\) is greater than the \(y -\) values of the neighboring points). The point \((-2,6)\) and \((2,0)\) is not. Wait, no, actually, the point \((-2,6)\) and if we check the other points:
The local maxima:
The function \(y = f(x)\): At \(x=-2\), \(y = 6\). For local minima:
A local minimum is a point where the function changes from decreasing to increasing. At \(x = 1\), the function value is \(y=-1\) (the point \((1, - 1)\)) since the function is decreasing before \(x = 1\) (in a small interval) and increasing after \(x = 1\) (in a small interval).

Answer:

Local maxima: \((-2,6)\)
Local minima: \((1,-1)\)