Let . Use a graphing utility to graph the functions and where . Identify the graphs of and below.

Question

Let  . Use a graphing utility to graph the functions  and  where  . Identify the graphs of  and  below. ​

Quizplus · Accepted Answer

The given functions are:
f(x) = 11eb0d3b_9811_158e_9790_0198fb59b3cb_TB1243_11 = x^3 - 5x
g(x) = 11eb0d3b_9811_158f_9790_694a405f6c3e_TB1243_11 = 3x^2 + 2x - 5
The derivative of f(x) is f'(x) = 11eb0d3b_9811_1591_9790_37294e4472a2_TB1243_11 = 3x^2 - 5 and the derivative of g(x) is g'(x) = 11eb0d3b_9811_1592_9790_f32686c61f09_TB1243_11 = 6x + 2.
The graphs of f(x) and g(x) will have critical points where their derivatives are zero. Therefore, we need to solve the equations f'(x) = 0 and g'(x) = 0:
f'(x) = 3x^2 - 5 = 0
=> x = +/-sqrt(5/3)
g'(x) = 6x + 2 = 0
=> x = -1/3
The critical points are (-sqrt(5/3), f(-sqrt(5/3))), (sqrt(5/3), f(sqrt(5/3))), and (-1/3, g(-1/3)).
Now we need to determine whether these critical points are local maxima or minima. We can use the second derivative test for this. The second derivative of f(x) is f''(x) = 11eb0d3b_9811_1593_9790_6124f6a8290c_TB1243_00 = 6x, and the second derivative of g(x) is g''(x) = 11eb0d3b_9811_1594_9790_f16ffdf1b201_TB1243_00 = 6.
At x = -sqrt(5/3), f''(-sqrt(5/3)) = -2*sqrt(5/3) < 0, so this critical point is a local maximum of f(x). At x = sqrt(5/3), f''(sqrt(5/3)) = 2*sqrt(5/3) > 0, so this critical point is a local minimum of f(x). At x = -1/3, g''(-1/3) = 6 > 0, so this critical point is a local minimum of g(x).
Therefore, the graph of f(x) has a local maximum at (-sqrt(5/3), f(-sqrt(5/3))), and a local minimum at (sqrt(5/3), f(sqrt(5/3))). The graph of g(x) has a local minimum at (-1/3, g(-1/3)).
Out of the given answer choices, only C matches this information about the local maxima and minima of the two functions. Therefore, the answer is C.

Let Use a Graphing Utility to Graph the Functions