What is the relative maximum and minimum of the function f in parenthesis x equals x3 plus 6x2 minus 36x?

To find the relative maximum and minimum of the function ( f(x) = x^3 + 6x^2 - 36x ), we first compute its derivative ( f'(x) = 3x^2 + 12x - 36 ). Setting the derivative equal to zero, we solve ( 3x^2 + 12x - 36 = 0 ), which simplifies to ( x^2 + 4x - 12 = 0 ). The roots of this equation are ( x = -6 ) and ( x = 2 ). Evaluating the second derivative ( f''(x) = 6x + 12 ), we find that ( f''(-6) < 0 ) indicates a relative maximum at ( x = -6 ), while ( f''(2) > 0 ) indicates a relative minimum at ( x = 2 ).