Let `f:[2,7]vec[0,oo)` be a continuous and differentiable function. Then show that `(f(7)-f(2))((f(7))^2+(f(2))^2+f(2)f(7))/3=5f^2(c)f^(prime)(c),` where `c in [2,7]dot`

Let f(x) be a twice-differentiable function and f''(0)=2. Then evaluate lim_(xto0) (2f(x)-3f(2x)+f(4x))/(x^(2)).

Let f(x) be a continuous and differentiable function such that f(x)=int_0^xsin(t^2-t+x)dt Then prove that f^('')(x)+f(x)=cosx^2+2xsinx^2

Let f:(0,oo)->R be a differentiable function such that f'(x)=2-f(x)/x for all x in (0,oo) and f(1)=1 , then

Let f: RvecR be a continuous and differentiable function such that (f(x^2+1))^(sqrt(x))=5forAAx in (0,oo), then the value of (f((16+y^2)/(y^2)))^(4/(sqrt(y)))fore a c hy in (0,oo) is equal to (a) 5 (b) 25 (c) 125 (d) 625

If the function f:[0,4]vecR is differentiable, the show that for a , b , in (0,4),f(4))^2-(f(0))^2=8f^(prime)(a)f(b)

Let f(x0 be a non-constant thrice differentiable function defined on (-oo,oo) such that f(x)=f(6-x)a n df^(prime)(0)=0=f^(prime)(x)^2=f(5)dot If n is the minimum number of roots of (f^(prime)(x)^2+f^(prime)(x)f^(x)=0 in the interval [0,6], then the value of n/2 is___

Let f:(-oo,oo)vec[0,oo) be a continuous function such that f(x+y)=f(x)+f(y)+f(x)f(y),AAx in Rdot Also f'(0)=1. Then [f(2)] equal ([dot] represents the greatest integer function ) 5 b. 6 c. 7 d. 8

Let f: (-5,5) rarr R be a differentiable function with f(4) = 1, f^(')(4) = 1, f(0) = -1 and f^(')(0) = 1 , If g(x) = f(2f^(2)(x)+2))^(2) , then - g^(')(0) equals

Let f(x)=|x-1|dot Then (a) f(x^2)=(f(x))^2 (b) f(x+y)=f(x)+f(y) (c) f(|x|)-|f(x)| (d) none of these

Let fa n dg be differentiable on [0,1] such that f(0)=2,g(0),f(1)=6a n dg(1)=2. Show that there exists c in (0,1) such that f^(prime)(c)=2g^(prime)(c)dot