Question

the graph of ( y = f(x) ) is shown to the right. identify the ( x )-coordinates of the points where ( f(x) ) has a local maximum. which of the following shows every ( x )-coordinate where ( f(x) ) has a local maximum? choose the correct answer below. a. ( a,f,h ) b. ( b,e,g ) c. ( a,c,h ) d. ( c,f )

Explanation:

Step1: Recall the definition of local maximum

A local maximum of a function \(y = f(x)\) occurs at a point \(x = c\) if \(f(c)\geq f(x)\) for all \(x\) in some open interval containing \(c\). Graphically, at a local maximum, the function changes from increasing to decreasing.

Step2: Analyze the given graph

Looking at the graph:

• At point \(x = a\), the function is at an endpoint. Endpoints are not considered for local maxima (by the standard definition of local maxima which requires an open - interval around the point).
• At point \(x = c\), the function changes from increasing (as we move from left to right towards \(c\)) to decreasing (as we move from left to right away from \(c\)).
• At point \(x = f\), the function changes from increasing (as we move from left to right towards \(f\)) to decreasing (as we move from left to right away from \(f\)).
• At point \(x = h\), it is an endpoint.
• Points \(b\), \(e\), \(g\) are local minima (the function changes from decreasing to increasing at these points).

Answer:

D. \(c,f\)