Question

math 104
3.4 second derivative test
3.4 second derivative test
question 1. consider the fourth degree polynomial function ( f(x) ) graphed below.
the graph of ( f(x) )
(a) identify the relative extrema of ( f(x) ).
(b) what is the concavity of the ( f(x) ) at the relative maximum?
(c) what is the concavity of the ( f(x) ) at the relative minima?

Explanation:

Step1: Recall the definition of relative extrema

Relative extrema are points where the function changes from increasing to decreasing (relative maximum) or from decreasing to increasing (relative minimum).

Step2: Analyze the graph

Looking at the graph of \(y = f(x)\), we can identify the points.

• For the relative maximum: The function has a "peak" at \(x = 0\).
• For the relative minima: The function has "valleys" at \(x=-2\) and \(x = 2\)

Answer:

The relative maximum is at \((0,0)\) and the relative minima are at \((-2,-4)\) and \((2,-4)\)