Find the relative maxima and relative minima, if any, of the function.

Question

F(X)= (1/3)x³ - x² - 3x + 3

relative maximum (x,y) = ( )

relative minimum (x,y) = ( )

Answer 1

f(x) = (1/3)x^3 -x^2 -3x + 3

take the derivative and set equal to zero

f'(x) = x^2 -2x -3 = 0

= (x-3)(x+1) = 0

x =-1, 3

f(-1) = y(-1) = (1/3)(-1)^3 -(-1)^2 -3(-1) +3 = -1/3 -1 +3 +3 = 4 2/3

f(3) =(1/3)(3)^3 - (3)^2 - 3(3) + 3 = 9 -9 - 9 + 3 = -6,

the points (-1, 4 2/3) and (3, -6) are local extrema; 4 2/3 > -6 so

(x,y) = (-1, 4 2/3) or (-1, 14/3) is a local or relative maximum

(x,y) = (3,-6) is a local or relative minimum