This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here's the solution for the first problem. Problem 1: Determine the critical points, their nature, and sketch the graph for y = (1)/(3)x^3 + (1)/(2)x^2 - 6x + 8. Step 1: Find the first derivative. To find the critical points, we first calculate the first derivative of the function y with respect to x. y' = (d)/(dx)((1)/(3)x^3 + (1)/(2)x^2 - 6x + 8) y' = (1)/(3)(3x^2) + (1)/(2)(2x) - 6 y' = x^2 + x - 6 Step 2: Determine the critical points. Set the first derivative equal to zero to find the x-coordinates of the critical points. x^2 + x - 6 = 0 Factor the quadratic equation: (x+3)(x-2) = 0 This gives two critical x-values: x = -3 or x = 2 Now, substitute these x-values back into the original function y = (1)/(3)x^3 + (1)/(2)x^2 - 6x + 8 to find the corresponding y-values. For x = -3: y = (1)/(3)(-3)^3 + (1)/(2)(-3)^2 - 6(-3) + 8 y = (1)/(3)(-27) + (1)/(2)(9) + 18 + 8 y = -9 + 4.5 + 26 y = 21.5 The first critical point is (-3, 21.5). For x = 2: y = (1)/(3)(2)^3 + (1)/(2)(2)^2 - 6(2) + 8 y = (1)/(3)(8) + (1)/(2)(4) - 12 + 8 y = (8)/(3) + 2 - 4 y = (8)/(3) - 2 y = (8)/(3) - (6)/(3) = (2)/(3) The second critical point is (2, (2)/(3)). The critical points are (-3, 21.5) and (2, (2)/(3)). Step 3: Determine the nature of the critical points. To determine if these points are local maxima or minima, we use the second derivative test. First, find the second derivative: y'' = (d)/(dx)(x^2 + x - 6) y'' = 2x + 1 Now, evaluate y'' at each critical x-value: For x = -3: y''(-3) = 2(-3) + 1 = -6 + 1 = -5 Since y''(-3) < 0, the critical point (-3, 21.5) is a local maximum. For x = 2: y''(2) = 2(2) + 1 = 4 + 1 = 5 Since y''(2) > 0, the critical point (2, (2)/(3)) is a local minimum. Step 4: Sketch the graph. The function is a cubic polynomial with a positive leading coefficient ((1)/(3)x^3). This means the graph will generally rise from left to right. • As x -, y -. • As x , y . The graph will increase until it reaches the local maximum at (-3, 21.5), then decrease until it reaches the local minimum at (2, (2)/(3)), and finally increase indefinitely. That's 2 down. 3 left today — send the next one.