Find the maximum and minimum values of the function f(x,y,z,t)=x+y+z+t subject to the constraint x^2+y^2+z^2+t^2=1

Question

Find the maximum and minimum values of the function f(x,y,z,t)=x+y+z+t subject to the constraint x^2+y^2+z^2+t^2=1 .

Maximum value is____ occurring at 1, points (positive integer or "infinitely many").

Minimum value is____ occurring at 1, points (positive integer or "infinitely many").

Answer 1

F(x,y,z,t) = x + y + z + t - λ(x² + y² + z² + t² - 1);

F_x = 1 - 2λx = 0; F_y = 1 - 2λy = 0; F_z = 1 - 2λz = 0; F_t = 1 - 2λt = 0; F_λ = - (x² + y² + z² + t² - 1) = 0

x = y = z = t = 1/(2λ); 1/λ² = 1; λ = ±1;

λ= 1; x = y = z = t = 1/2; f(1/2, 1/2,1/2, 1/2) = 2 (max)

λ = - 1; x = y = z = t = -1/2; f(-1/2,-1/2,-1/2,-1/2) = - 2 (min)