Find the largest natural number a for which the maximum value of `f(x)=a-1+2x-x^2` is smaller than the minimum value of `g(x)=x^2-2a x+10-2adot`

Find the minimum value of |x|+|x+1/2|+|x-3|+|x-5/2|dot

Find the minimum value of 2^("sin" x) + 2^("cos" x)

For all real values of x, the minimum value of (1-x+x^(2))/(1+x+x^(2)) is

f(x)=((x-2)(x-1))/(x-3), forall xgt3 . The minimum value of f(x) is equal to

find the maximum value of f(x) = (sin^(-1) (sin x))^(2) - sin^(-1) (sin x)

Let f(x)=x^2-2x-1 AA x in R Let f:(-oo, a]->[b, oo) , where a is the largest real number for which f(x) is bijective. If f : R->R , g(x) = f(x) + 3x-1 , then the least value of function y = g(|x|) is

If f(x)=sqrt(4-x^2)+sqrt(x^2-1) , then the maximum value of (f(x))^2 is ____________

If f(x)=sqrt(4-x^2)+sqrt(x^2-1) , then the maximum value of (f(x))^2 is ____________

Find the largest possible domain for the real valued function given f(x) = (sqrt(9-x^2) )/(sqrt(x^2 - 1) )

Find the absolute maximum value and the absolute minimum value of the functions in the given intervals: f(x) = (x-1)^(2) +3, x in [-3 ,1)