Let `f(x)=(x-a)(x-b)(x-c), a lt b lt c.` Show that `f'(x)=0` has two roots one belonging to (a, b) and other belonging to (b, c).

If f(x)= (x-a)^(2) (x-b)^(2) then find f(a+b)

If a lt b lt c lt d then show that (x-a)(x-c)+3(x-b)(x-d)=0 has real and distinct roots.

Let f(x) = (g(x))/(h(x)) , where g and h are continuous functions on the open interval (a, b). Which of the following statements is true for a lt x lt b ?

f(x)=(x-a)^(2)+(x-b)^(2)+(x-c)^(2) has minimum value at x = …………..

Let f:(0,1)->R be defined by f(x)=(b-x)/(1-bx) , where b is constant such that 0 ltb lt 1 .then ,(a)f is not invertible on (0,1) (b) f ≠ f − 1 on (0,1) and f ' ( b ) = 1/ f ' ( 0 ) (c) f = f − 1 on (0,1) and f ' ( b ) = 1/ f ' ( 0 ) (d) f − 1 is differentiable on (0,1)

Let a , b , c be nonzero real numbers such that int_0^1(1+cos^8x)(a x^2+b x+c)dx =int_0^2(1+cos^8x)(a x^2+b x+c)dx=0 Then show that the equation a x^2+b x+c=0 will have one root between 0 and 1 and other root between 1 and 2.

If a x^2+(b-c)x+a-b-c=0 has unequal real roots for all c in R ,then

Find f'(x) if f(x)= (sin x)^(sin x) for all 0 lt x lt pi .

Let f be a function defined on [a,b] such that f'(x)gt 0 , for all x in (a, b) . Then prove that f is an increasing function on (a, b).