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.
ScanSolve AI Answer
7. (ii) Step 1: Find (dx)/(d) and (dy)/(d). Given x = 2 and y = 2. (dx)/(d) = (d)/(d)( 2) = 2 2 (dy)/(d) = (d)/(d)( 2) = -2 2 Step 2: Find (dy)/(dx) using the chain rule. (dy)/(dx) = (dy/d)/(dx/d) = (-2 2)/(2 2) (dy)/(dx) = -( 2)/( 2) = - 2 Step 3: Find (d^2y)/(dx^2) using the chain rule. (d^2y)/(dx^2) = (d)/(d)((dy)/(dx)) · (1)/(dx/d) First, find (d)/(d)((dy)/(dx)): (d)/(d)(- 2) = -^2(2) · (d)/(d)(2) = -^2(2) · 2 = -2 ^2(2) Now substitute this back into the formula for (d^2y)/(dx^2): (d^2y)/(dx^2) = (-2 ^2(2)) · (1)/(2 2) (d^2y)/(dx^2) = (-2 ^2(2))/(2 2) Since 2 = (1)/( 2): (d^2y)/(dx^2) = (-1)/(^2(2) 2) (d^2y)/(dx^2) = (-1)/(^3(2)) (d^2y)/(dx^2) = -^3(2) 7. (iii) Step 1: Find the first derivative of the function f(x). Given f(x) = (2x-5)^3 + x. f'(x) = (d)/(dx)((2x-5)^3) + (d)/(dx)(x) Using the chain rule for (2x-5)^3: (d)/(dx)((2x-5)^3) = 3(2x-5)^2 · (d)/(dx)(2x-5) = 3(2x-5)^2 · 2 = 6(2x-5)^2 So, the derivative f'(x) is: f'(x) = 6(2x-5)^2 + 1 Step 2: Analyze the sign of f'(x). For any real number x, the term (2x-5)^2 is always greater than or equal to zero because it is a square. (2x-5)^2 0 Multiplying by 6, the term remains non-negative: 6(2x-5)^2 0 Adding 1 to this expression: 6(2x-5)^2 + 1 1 Therefore, f'(x) 1 for all x R. Step 3: Conclude that the function is increasing. Since f'(x) 1, it means f'(x) > 0 for all x R. A function is increasing if its first derivative is always positive. Thus, the function f(x) = (2x-5)^3 + x is an increasing function. Send me the next one 📸