Use differentiation to determine the point(s) of inflection of the function: y = x³ + 2x² − 5x − 6

Step 1: Find the first derivative of the function. The given function is y = x^3 + 2x^2 - 5x - 6. Differentiate y with respect to x: (dy)/(dx) = (d)/(dx)(x^3) + (d)/(dx)(2x^2) - (d)/(dx)(5x) - (d)/(dx)(6) (dy)/(dx) = 3x^2 + 4x - 5 Step 2: Find the second derivative of the function. Differentiate (dy)/(dx) with respect to x: (d^2y)/(dx^2) = (d)/(dx)(3x^2) + (d)/(dx)(4x) - (d)/(dx)(5) (d^2y)/(dx^2) = 6x + 4 Step 3: Set the second derivative to zero to find potential points of inflection. Points of inflection occur where the concavity changes, which typically happens when (d^2y)/(dx^2) = 0. 6x + 4 = 0 6x = -4 x = -(4)/(6) x = -(2)/(3) Step 4: Verify that the concavity changes at x = -(2)/(3). We can check the sign of y'' for values of x less than and greater than -(2)/(3). If x < -(2)/(3) (e.g., x = -1): y'' = 6(-1) + 4 = -6 + 4 = -2 < 0 (concave down). If x > -(2)/(3) (e.g., x = 0): y'' = 6(0) + 4 = 4 > 0 (concave up). Since the sign of y'' changes at x = -(2)/(3), this is indeed an inflection point. Step 5: Substitute the x-value back into the original function to find the corresponding y-coordinate. Substitute x = -(2)/(3) into y = x^3 + 2x^2 - 5x - 6: y = (-(2)/(3))^3 + 2(-(2)/(3))^2 - 5(-(2)/(3)) - 6 y = -(8)/(27) + 2((4)/(9)) + (10)/(3) - 6 y = -(8)/(27) + (8)/(9) + (10)/(3) - 6 To combine these, find a common denominator, which is 27: y = -(8)/(27) + (8 × 3)/(9 × 3) + (10 × 9)/(3 × 9) - (6 × 27)/(1 × 27) y = -(8)/(27) + (24)/(27) + (90)/(27) - (162)/(27) y = (-8 + 24 + 90 - 162)/(27) y = (16 + 90 - 162)/(27) y = (106 - 162)/(27) y = -(56)/(27) The point of inflection is (-(2)/(3), -(56)/(27)). 3 done, 2 left today. You're making progress.