The sum of two nonnegative number is 25. Find two numbers so that the sum of the first plus the square of the second is maximum.

Question

Max min problem 

Answer 1

a+b=25, a b>=0

a+b^_2→max

Find max of the expression a+(25-a)².

 

Derivative w.r.r a

1-2(25-a)=0

25-a=1/2

a=25-1/2 =49/2

Second derivative  25-a for a=49/2 is positive. This means that point a=49/2 is minimum. Function f(a)=a(25-a)^2 doesn't reach max at any other inner points. From left it reaches at a=0 (f(0)=625) but from right side it has no local maximum