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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Fresh day stephennaomi63, let's solve. You've asked to solve the questions from the image. I've already completed question 4.a, 4.b, 4.c, and 4.d. I will now proceed with question 1.a. Question 1.a asks: i) What do you mean by the minimum point of a function? ii) The function y = 2x^3 + ax^2 + bx where a and b are constant, has critical points at x=2 and x=1. Find: (i) The values of a and b. (ii) The values of x for which y is minimum. (iii) The minimum value of y. Step 1: Define the minimum point of a function. A minimum point of a function is a point where the function's value is less than or equal to the values at all nearby points within its domain. At a local minimum, the first derivative of the function is zero (if differentiable), and the second derivative is positive. Step 2: Find the values of a and b. Given the function y = 2x^3 + ax^2 + bx. Critical points occur where the first derivative y' is equal to zero. First, find the first derivative y': y' = (d)/(dx)(2x^3 + ax^2 + bx) y' = 6x^2 + 2ax + b The critical points are given as x=1 and x=2. This means y'=0 at these points. For x=1: 6(1)^2 + 2a(1) + b = 0 6 + 2a + b = 0 2a + b = -6 (Equation 1) For x=2: 6(2)^2 + 2a(2) + b = 0 6(4) + 4a + b = 0 24 + 4a + b = 0 4a + b = -24 (Equation 2) Now, solve the system of linear equations for a and b. Subtract Equation 1 from Equation 2: (4a + b) - (2a + b) = -24 - (-6) 2a = -18 a = (-18)/(2) a = -9 Substitute the value of a=-9 into Equation 1: 2(-9) + b = -6 -18 + b = -6 b = -6 + 18 b = 12 The values of a and b are a = -9, b = 12. What's next? Send 'em.