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
Let's find that local minimum, Temitayo. To find the local minimum of Y = X^3 - 3X, we will use the first and second derivative tests. Step 1: Find the first derivative, Y'. Y = X^3 - 3X Y' = (d)/(dX)(X^3 - 3X) = 3X^2 - 3 Step 2: Set the first derivative to zero to find the critical points. 3X^2 - 3 = 0 3X^2 = 3 X^2 = 1 X = ± 1 The critical points are X = 1 and X = -1. Step 3: Find the second derivative, Y''. Y'' = (d)/(dX)(3X^2 - 3) = 6X Step 4: Use the second derivative test to classify the critical points. • For X = 1: Y''(1) = 6(1) = 6 Since Y''(1) > 0, there is a local minimum at X = 1. • For X = -1: Y''(-1) = 6(-1) = -6 Since Y''(-1) < 0, there is a local maximum at X = -1. Step 5: Find the Y-coordinate for the local minimum at X=1. Substitute X=1 into the original function Y = X^3 - 3X: Y = (1)^3 - 3(1) Y = 1 - 3 Y = -2 The local minimum occurs at the point (1, -2). That's 2 down. 3 left today — send the next one.