Check the boundary conditions. For example, in the optimization problem where Y = X^2 and the boundaries are -1 and +3, check the Y values at X = -1 and X = +3. Y(-1) = (-1)^2 = 1 and Y(+3) = 3^2 = 9. So, a possible minimum is 1 and possible maximum is 9. Because the function is nonlinear, it is possible there is another minimum or maximum between X = -1 and X = +3.
Find the derivative of the function to locate extrema -- places where the function maximizes or minimizes. The derivative of a function is another function that describes how the original function changes. At the place where the derivative is zero, the original function has stopped changing -- because it has reached a maximum or minimum and is changing direction. To find the derivative of a polynomial, change each term according to this rule: aX^n becomes anX^(n-1). For example, the derivative of 2X^3 + 5X^2 - 3X + 17 is 6X^2 +10X - 3. The 17 vanishes because it is a constant and never changes. Derivatives describe change.
Compare the boundary conditions to the extrema. The derivative of X^2 is 2X. If we set the derivative to zero -- to find the place where the curve changes direction -- we get the equation 2X = 0. The solution is X = 0, so the curve changes direction when X = 0. Y(0 ) = 0, so this point is a minimum -- it is smaller than both the boundary values. The optimum values of Y = X^2 between are 0 and 9 -- when X = 0 and X = +3.