Let `f:(a , b)->R` is a differentiable function such that `(lim)_(x->a^+)f^2(x)=0,("lim")_(x->b^-)f^2(x)=e-1` and `2f(x)f^(prime)(x)-f^2(x)geq1` for all `x in (a , b)` then value of `(b-a)` can be (a) 0 (b) `1/2` (c) 1 (d) 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

If the function f(x) satisfies (lim)_(x->1)(f(x)-2)/(x^2-1)=pi, evaluate (lim)_(x->1)f(x)

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

Let f(x) be a twice-differentiable function and f"(0)=2. The evaluate: ("lim")_(xvec0)(2f(x)-3f(2x)+f(4x))/(x^2)

Suppose f(x)=e^(a x)+e^(b x) , where a!=b , and that f''(x)-2f^(prime)(x)-15f(x)=0 for all xdot Then the value of (|a b|)/3 is ___

Let f(x) be a function such that ("lim")_(xvec0)(f(x))/x=1 and ("lim")_(xvec0)(x(1+acosx)-bsinx)/((f(x))^3)=1 , then b-3a is equal to

Let f:[0,2]->R be a function which is continuous on [0,2] and is differentiable on (0,2) with f(0)=1 L e t :F(x)=int_0^(x^2)f(sqrt(t))dtforx in [0,2]dotIfF^(prime)(x)=f^(prime)(x) . for all x in (0,2), then F(2) equals (a) e^2-1 (b) e^4-1 (c) e-1 (d) e^4

If f(x) is a differentiable real valued function such that f(0)=0 and f\'(x)+2f(x) le 1 , then (A) f(x) gt 1/2 (B) f(x) ge 0 (C) f(x) le 1/2 (D) none of these

If f(x) is differentiable and strictly increasing function, then the value of ("lim")_(xvec0)(f(x^2)-f(x))/(f(x)-f(0)) is 1 (b) 0 (c) -1 (d) 2

The function f: RvecR satisfies f(x^2)dotf^(x)=f^(prime)(x)dotf^(prime)(x^2) for all real xdot Given that f(1)=1 and f^(1)=8 , then the value of f^(prime)(1)+f^(1) is 2 b. 4 c. 6 d. 8