Find all the points of local maxima and minima and the corresponding maximum and minimum values of the function `f(x)=2x^3-21x^2+36 x-20.`

Explanation

Given, ` f(x) = x^(3) - 21 x^(2) + 36 x - 20`
` :. " " f'(x) = 6x^(2) - 42x + 36`
` :. " " f''(x) = 12x - 42`
Taking ` f'(x) = 0`
` :. 6x^(2) - 42x + 36 = 0`
` rArr 6 (x^(2) - 7x + 6) = 0`
` rArr x^(2) - 7x + 6 = 0`
` rArr (x - 6)(x-1) = 0`
` :. x = 6 or x = 1`.
Now ` f'' (x = 6) = 12 xx 6 - 42 = 72 - 42 = 30 gt 0`
` :. ` f (x) has minima at x = 6
Min. value fo f (x) at x = 6 is
` f(6) = 2 (6)^(3) - 21 (6)^(2) + 36 xx 6 - 20`
` = 2 xx 216 - 21 xx 36 + 36 xx 6 - 20`
` = 432 - 756 + 216 - 20`
` = - 128`
` f'' (x = 1) = 12 * 1 - 42 = 12 - 42 = - 30 lt 0 `
` :. ` f (x) has maxima at x = 1 .
` f(1) = 2 * 1^(3) - 21 * 1^(2) + 36 * 1 - 20 `
` = 2 - 21 + 36 - 20`
` = 38 - 41 = - 3`