Given f'(x) as a piecewise { f'(x) = x+2 for x<=-1, f'(x) = x^2 for x>= -1, determine all possibilities for the equation of the function f.

Answer 1

{f(x) = (x + 2)²/2 + C₁ for x ≤ - 1, f(x) = x³/3 + C₂ for x ≥ -1} only at x = - 1 we have to have equal values. So, 1/2 + C₁ = -1/3 + C₂; C₂ = C₁ + 5/6. {f(x) = (x + 2)²/2 + C₁ for x ≤ - 1, f(x) = x³/3 + C₁ + 5/6 for x ≥ -1}