For a twice differentiable function `f(x),g(x)` is defined as `g(x)=f^(prime)(x)^2+f^(prime)(x)f(x)on[a , e]dot` If for `a

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: R->R be a differentiable function with f(0)=1 and satisfying the equation f(x+y)=f(x)f^(prime)(y)+f^(prime)(x)f(y) for all x ,\ y in R . Then, the value of (log)_e(f(4)) is _______

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

If function f satisfies the relation f(x)*f^(prime)(-x)=f(-x)*f^(prime)(x) for all x ,

Let f(x)a n dg(x) be two differentiable functions in Ra n df(2)=8,g(2)=0,f(4)=10 ,a n dg(4)=8. Then prove that g^(prime)(x)=4f^(prime)(x) for at least one x in (2,4)dot

Let f(x) be a non-constant twice differentiable function defined on (-oo,oo) such that f(x)=f(1-x)a n df^(prime)(1/4)=0. Then (a) f^(prime)(x) vanishes at least twice on [0,1] (b) f^(prime)(1/2)=0 (c) int_(-1/2)^(1/2)f(x+1/2)sinxdx=0 (d) int_(-1/2)^(1/2)f(t)e^(sinpit)dt=int_(1/2)^1f(1-t)e^(sinpit)dt

Let f(x)a n dg(x) be differentiable functions such that f^(prime)(x)g(x)!=f(x)g^(prime)(x) for any real xdot Show that between any two real solution of f(x)=0, there is at least one real solution of g(x)=0.

Let f: R->R and g: R->R be two non-constant differentiable functions. If f^(prime)(x)=(e^((f(x)-g(x))))g^(prime)(x) for all x in R , and f(1)=g(2)=1 , then which of the following statement(s) is (are) TRUE? f(2) 1-(log)_e2 (c) g(1)>1-(log)_e2 (d) g(1)<1-(log)_e2

Let f(x)a n dg(x) be two functions which are defined and differentiable for all xgeqx_0dot If f(x_0)=g(x_0)a n df^(prime)(x)>g^(prime)(x) for all x > x_0, then prove that f(x)>g(x) for all x > x_0dot

Evaluate: ifintg(x)dx=g(x),t h e nintg(x){f(x)+f^(prime)(x)}dx