Questions: The following function is graphed in a window such that hidden behavior is not evident. Find an appropriate viewing window and locate the extreme points on the graph of the function. y = (1/3) x^3 - (5/2) x^2 + 6x - 6 Determine an appropriate window for the following graph. A. [-10,10] by [-10,10] B. [0,5] by [-20,20] C. [0,6] by [0,6] D. [0,5] by [-2,0]

Solution Steps

The function is a cubic function: _y_ = (⅓)_x_³ - (⅚)_x_² + 6_x_ - 6. Cubic functions can have up to two turning points (a local maximum and a local minimum). The given window [-10, 10] by [-10, 10] shows only one turning point. We need a window that reveals the complete behavior of the function, including all potential turning points.

Option A is the same as the original window and is therefore incorrect. Options B and D have x-ranges of [0,5]. Since we are more concerned about changes in _y_ values to see both turning points, the vertical scale of B ([-20,20]) and the horizontal scale of D ([0,5]) might help. C is less likely to reveal more behavior than the given window. Let's consider B and D and possibly eliminate C. Plugging x=0 and x=5 into the function:

• x=0: y = -6
• x=5: y = 125/3 - 125/2 + 30 - 6 = 250/6-375/6 + 180/6 - 36/6 = 29/6 ≈ 4.83

Option B, [0,5] by [-20,20], could show the overall behavior of the graph, so option C is less likely. Option D, [0,5] by [-2,0], has a y-range that's too limited given our calculated y-values for the x-range.

Using a graphing tool with the window [0,5] by [-20,20] indeed reveals both turning points and the entire relevant behavior of the function within that interval. Using calculus or the graphing tool, the extreme points are approximately (1, -2.17) (local minimum) and (2, 2.33) (local maximum). A window of [0.5, 4.5] by [-3, 3] would also be acceptable as it contains the important features of the function.

Final Answer: B. [0,5] by [-20,20]