If `f:[-5,5]vecR` is differentiable function and if`f^(prime)(x)` does not vanish anywhere, then prove that `f(-5)!=f(5)dot`

If f: [-5, 5] rarr R is a differentiable function and if f'(x) does not vanish anywhere, then prove that f(-5) ne f(5)

Let f(x+y)=f(x)+f(y)+2x y-1 for all real xa n dy and f(x) be a differentiable function. If f^(prime)(0)=cosalpha, the prove that f(x)>0AAx in Rdot

Let f be a differentiable function satisfying [f(x)]^(n)=f(nx)" for all "x inR. Then, f'(x)f(nx)

A function f: R rarr R satisfies the equation f(x+ y)= f(x).f(y) "for all" x, y in R, f(x) ne 0 . Suppose that the function is differentiable at x=0 and f'(0)=2, then prove that f'(x)= 2f(x)

If f(x)= int_(0)^(x)(f(t))^(2) dt, f:R rarr R be differentiable function and f(g(x)) is differentiable at x=a , then

If y=f(x) is an odd differentiable function defined on (-oo,oo) such that f^(prime)(3)=-2,t h e n|f^(prime)(-3)| equals_________.

Let f be differentiable function satisfying f((x)/(y))=f(x) - f(y)"for all" x, y gt 0 . If f'(1) = 1, then f(x) is

Let f be a twice differentiable function such that f''(x)=-f(x),a n df^(prime)(x)=g(x),h(x)=[f(x)]^2+[g(x)]^2dot Find h(10)ifh(5)=11

Find the derivative of the function f(x)=2x^(2)+3x-5" at "x=-1 . Also prove that f'(0)+3f'(-1)=0 .

If f(x)=(a x^2+b)^3, then find the function g such that f(g(x))=g(f(x))dot