Find the absolute maximum and absolute minimum values of fx

Find the absolute maximum and absolute minimum values of f(x) = x2/3 - 2x1/3 on the interval -8 x 27. Find the absolute maximum and absolute minimum values of f(x) = x/ex on the interval 0 x 2.

Solution

a) f(x) = x^2/3 - 2x^1/3 f\'(x) = 2/3x^-1/3 - 2/3x^-2/3 put f\'(x) = 0 2/3x^-1/3 - 2/3x^-2/3 = 0 x^-1/3 = x^-2/3 x^2/3 = x^1/3 cubcing both sides x^2 = x x(x-1) = 0 x = 0,1 f\'\'(x) = -2/9x^-4/3 + 4/9x^-5/3 f\'\'(0) = 0....inflection point at x = 0 f\'\'(1) = -2/9+4/9 = 2/9.......absolute minima occurs at x = 1 f(1) = 1-2 = -1.........absolute minimum b) f(x) = x/e^x = xe^-x f\'(x) = e^-x - xe^-x = 0 e^-x(1-x) = 0 x = 1 f\'\'(x) = -e^-x - e^-x + xe^-x = -2e^-x + xe^-x f\'\'(1) = -2/e + 1/e = -1/e.....absolute maxima at x =1 f(1) = 1/e...absolute maximum value