Question

construct arguments sheena says that in the graph of ( f(x) ) shown below, the graph has relative maxima at ( b ) and ( g ), and a relative minimum at ( a ). is she correct? explain. sheena is select choice, while ( b ) and ( g ) select choice relative maxima, ( a ) select choice a minimum, whereas / select choice.

Explanation:

Step1: Recall relative max/min definitions

A relative maximum of a function \( f(x) \) at a point \( c \) means \( f(c) \geq f(x) \) for \( x \) near \( c \). A relative minimum means \( f(c) \leq f(x) \) for \( x \) near \( c \).

Step2: Analyze point B

At point B, the graph peaks around that \( x \)-value (near \( x = -4 \)), so \( f(B) \) is greater than nearby points. So B is a relative maximum.

Step3: Analyze point G

At point G (near \( x = 2 \)), the graph peaks around that \( x \)-value, so \( f(G) \) is greater than nearby points. So G is a relative maximum.

Step4: Analyze point A

Point A is on the \( x \)-axis, but as \( x \) approaches A from the left, the function values go to \( -\infty \), and from the right, the function increases from A. But a relative minimum requires the function to be less than or equal to nearby points. However, since the left side goes to \( -\infty \), A is not a relative minimum (relative minima occur at points like D or H/J, where the function is lower than nearby points). Also, Sheena's claim about A is wrong.

Answer:

Sheena is incorrect. While B and G are relative maxima (since \( f(B) \geq f(x) \) for \( x \) near \( B \) and \( f(G) \geq f(x) \) for \( x \) near \( G \)), A is not a relative minimum. A relative minimum occurs at a point where the function value is less than or equal to nearby points; at A, as \( x \) approaches from the left, \( f(x) \to -\infty \), so A is not a relative minimum (points like D, H, or J are relative minima as they are lower than their immediate neighbors).