To compute some values by Xcas

If you compute the following values by Xcas, then you should execute commands below. See also this page.

Case 1. One computes any maximum values for given functions with 1 parameter, for example \( y = x^2 - 4x + 5 \textrm{ for } 37/19 \leq x \leq 35/17. \) Then, he should execute as follows. The answer must be \( 290/289. \)

maximize(x^2 - 4*x + 5, x=37/19..35/17)

Case 2. One computes any minimum values for given functions with 2 parameters, for example \( z = x^3 + y^3 - xy \textrm{ for } x > 0 \textrm{ and } y > 0. \) Then, he should execute as follows. The answer must be \( -1/27. \)

minimize(x^3 + y^3 - x*y, [x > 0, y > 0], [x, y])

Case 3. One computes any maximum values for given 2-parameter-functions with 1 bounded condition, so-called Lagrange Multiplier method, for example \( z = x/2 + y/3 + \sqrt{1 - (x^2 / 4) - (y^2 / 9)} \textrm{ with } x + y = 5\sqrt{3}/3. \) Then, he should execute as follows. The answer must be \( (10\sqrt{3} - 4)/9. \)

maximize((x/2) + (y/3) + sqrt(1 - (x^2 / 4) - sqrt(y^2/9)), x + y = 5*sqrt(3)/3, [x, y])

Case 4. One differentiates any implicit functions such as \( y \ln x = x \ln y \ (x > 0 \textrm{ and } y > 0) \). Then, he should execute as follows. The answer must be \( (dy / dx =) (y^2 - xy \ln y)/(x^2 - xy \ln x). \)

implicitdiff(y*ln(x) = x*ln(y), y, x)