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 are the solutions to Question 7: Given the function f(x) = -2x^3 - 3x^2. 7.1 To determine the x-intercepts, we set f(x) = 0. -2x^3 - 3x^2 = 0 Factor out the common term x^2: -x^2(2x + 3) = 0 This gives two possible solutions: -x^2 = 0 x = 0 2x + 3 = 0 2x = -3 x = -(3)/(2) The x-intercepts are (0, 0) and (-(3)/(2), 0). 7.2 To determine the y-intercept, we set x = 0. f(0) = -2(0)^3 - 3(0)^2 f(0) = 0 - 0 f(0) = 0 The y-intercept is (0, 0). 7.3 To determine the coordinates of the turning points, we first find the first derivative f'(x) and set it to zero. Step 1: Find the derivative f'(x). f'(x) = (d)/(dx)(-2x^3 - 3x^2) f'(x) = -2(3)x^3-1 - 3(2)x^2-1 f'(x) = -6x^2 - 6x Step 2: Set f'(x) = 0 to find the x-coordinates of the turning points. -6x^2 - 6x = 0 Factor out -6x: -6x(x + 1) = 0 This gives two possible solutions for x: -6x = 0 x = 0 x + 1 = 0 x = -1 Step 3: Substitute these x-values back into the original function f(x) to find the corresponding y-coordinates. For x = 0: f(0) = -2(0)^3 - 3(0)^2 = 0 Turning point 1: (0, 0) For x = -1: f(-1) = -2(-1)^3 - 3(-1)^2 f(-1) = -2(-1) - 3(1) f(-1) = 2 - 3 f(-1) = -1 Turning point 2: (-1, -1) The coordinates of the turning points are (0, 0) and (-1, -1). 7.4 To sketch the graph of f(x) = -2x^3 - 3x^2, we use the information gathered: • x-intercepts: (0, 0) and (-(3)/(2), 0) (or (-1.5, 0)) • y-intercept: (0, 0) • Turning points: (0, 0) (local maximum) and (-1, -1) (local minimum) The function is a cubic with a negative leading coefficient, so it generally starts from the top-left, goes down, turns up, then turns down to the bottom-right. The graph will: 1. Start from the top-left. 2. Pass through the x-intercept (-(3)/(2), 0). 3. Continue downwards to reach the local minimum at (-1, -1). 4. Turn upwards from the local minimum. 5. Reach the local maximum at (0, 0), which is also an x-intercept and the y-intercept. 6. Turn downwards from the local maximum and continue to the bottom-right. ` ^ y | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |