The maximum and minimum values of y occur when x^3 equals 8 ± 2√14. The equation defining y is y^3 = 6xy - x^3 - 1. To find these extrema, the first derivative dy/dx is set to zero, leading to the relationship y = x^2/2. Substituting the x values derived from the cubic equation into this relationship yields the corresponding y values, which can then be analyzed using the second derivative test to confirm their nature as maxima or minima.
PREREQUISITESStudents studying calculus, particularly those focusing on optimization problems, as well as educators seeking to enhance their teaching methods in differential calculus.
Liang Wei said:can I use the sign test to find the maximum and minimum values from this kind of equation?
Oh that one, for that you need to know the interval, and it requires you to solve for x.Liang Wei said:Sign test is a test which you use a value which lies inside the interval of a variable,example x then substitute it into dy/dx to find the sign and we can find the maxima and minima.
but anyways I manage to prove it by using 2nd derivative but not sure correct or wrong.