Question

a student sketched a graph of ( g(x)=x^{3}+6x^{2}+8x ) : the function ( g(x) ) has a relative of at approximately ( x=-3.15 ). the function ( g(x) ) has a relative of at approximately ( x=-0.85 ).

Explanation:

Step1: Analyze the graph at \(x = - 3.15\)

At \(x=-3.15\), the function \(g(x)\) changes from increasing to decreasing. According to the definition of relative extrema, when a function changes from increasing to decreasing, it has a relative maximum.

Step2: Analyze the graph at \(x=-0.85\)

At \(x =-0.85\), the function \(g(x)\) changes from decreasing to increasing. According to the definition of relative extrema, when a function changes from decreasing to increasing, it has a relative minimum.

Answer:

The function \(g(x)\) has a relative maximum at approximately \(x=-3.15\). The function \(g(x)\) has a relative minimum at approximately \(x =-0.85\).