If `a,b in R and a ltb,` then prove that there exists at least one real number `c in(a,b)"such that"(b^(2)+a^(2))/(4c^(2))=(c)/(a+b).`

If f(x) is differentiate in [a,b], then prove that there exists at least one c in (a,b)"such that"(a^(2)-b^(2))f'(c)=2c(f(a)-f(b)).

If f(x) is continuous in [a , b] and differentiable in (a,b), then prove that there exists at least one c in (a , b) such that (f^(prime)(c))/(3c^2)=(f(b)-f(a))/(b^3-a^3)

If f(x) is continuous in [a , b] and differentiable in (a,b), then prove that there exists at least one c in (a , b) such that (f^(prime)(c))/(3c^2)=(f(b)-f(a))/(b^3-a^3)

If A=2B , then prove that either c=b or a^2 = b(c+b)

If b^2<2a c , then prove that a x^3+b x^2+c x+d=0 has exactly one real root.

If (x) is differentiable in [a,b] such that f(a)=2,f(b)=6, then there exists at least one c,a ltcleb, such that (b^(3)-a^(3))f'(c)=

If a, b, c are three distinct positive real numbers, then the least value of ((1+a+a^(2))(1+b+b^(2))(1+c+c^(2)))/(abc) , is

If a , b ,c are three distinct positive real numbers in G.P., then prove that c^2+2a b >3ac

If a,b,c are real numbers such that 3(a^(2)+b^(2)+c^(2)+1)=2(a+b+c+ab+bc+ca) , than a,b,c are in

If a, b and c are positive real numbers such that aleblec, then (a^(2)+b^(2)+c^(2))/(a+b+c) lies in the interval