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
Here's whether each function is increasing: a) f(x) = x Step 1: Find the derivative of the function. f'(x) = (d)/(dx)(x) = 1 Step 2: Analyze the derivative. Since f'(x) = 1 > 0 for all real values of x, the function f(x) = x is always increasing. Yes, x is increasing. b) f(x) = x^3 Step 1: Find the derivative of the function. f'(x) = (d)/(dx)(x^3) = 3x^2 Step 2: Analyze the derivative. Since x^2 0 for all real values of x, 3x^2 0 for all x. The derivative is zero only at x=0, but it is positive for all other x. Therefore, the function f(x) = x^3 is always increasing. Yes, x^3 is increasing. c) f(x) = |x| Step 1: Define the function piecewise and find its derivative. f(x) = x & if x 0 \\ -x & if x < 0 The derivative is: f'(x) = 1 & if x > 0 \\ -1 & if x < 0 Step 2: Analyze the derivative. For x > 0, f'(x) = 1 > 0, so f(x) is increasing. For x < 0, f'(x) = -1 < 0, so f(x) is decreasing. Since the function is decreasing for x < 0, it is not an increasing function over its entire domain. No, |x| is not increasing. That's 2 down. 3 left today — send the next one.