Let `f(x)=e^(2x)` and `b=a+hdot` If there exists a real number `theta in (0,1)` such that `varphi^(prime)(a+thetah)=0` and `(e^(2h)-1-2h-2h^2)/(h^3)-A e^(bthetah),` then

Let f(x)=x^4-6x^3+12 x^2-8x+3. If Rolles theorem is applicable to varphi(x) on [2,2+h] and there exist c in (2,2+h) such that varphi^(prime)(c)=0 and (f(2+h)-f(2))/(h^3)=g(c), then

Let f:[1/2,1]->R (the set of all real numbers) be a positive, non-constant, and differentiable function such that f^(prime)(x)<2f(x)a n df(1/2)=1 . Then the value of int_(1/2)^1f(x)dx lies in the interval (a) (2e-1,2e) (b) (3-1,2e-1) (c) ((e-1)/2,e-1) (d) (0,(e-1)/2)

Let f:[1/2,1]->R (the set of all real numbers) be a positive, non-constant, and differentiable function such that f^(prime)(x)<2f(x))a n df(1/2)=1 . Then the value of int_(1/2)^1f(x)dx lies in the interval (a) (2e-1,2e) (b) (3-1,2e-1) ((e-1)/2,e-1) (d) (0,(e-1)/2)

If point E(5, h) is on the line that contains A(0,1) and B(-2, -1) , what is the value of h?

Let f: RvecR be such that f(a)=1,f^(prime)(a)=2. Then (lim)_(xvec0)((f^2(a+x))/(f(a)))^(1//x) is a. e^2 b. e^4 c. e^(-4) d. 1//e

Evaluate lim_(hto0) (log_(e)(1+2h)-2log_(e)(1+h))/(h^(2)).

Let a=int_0^(log2) (2e^(3x)+e^(2x)-1)/(e^(3x)+e^(2x)-e^x+1)dx , then 4e^a =

f:(2,3)vec(0,1)d efin e db yf(x)=x-[x],w h e r e[dot] represents the greatest integer function.

If f(x)={e^(cosxsinx ,for|x|lt=2)2,ot h e r w i s e ,t h e nint_(-2)^3f(x)dx= 0 (b) 1 (c) 2 (d) 3

The condition that one of the straight lines given by the equation a x^2+2h x y+b y^2=0 may coincide with one of those given by the equation a^(prime)x^2+2h^(prime)x y+b^(prime)y^2=0 is (a b^(prime)-a^(prime)b)^2=4(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (a b^(prime)-a^(prime)b)^2=(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (h a^(prime)-h^(prime)a)^2=4(a b^(prime)-a^(prime)b)(b h^(prime)-b^(prime)h) (b h^(prime)-b^(prime)h)^2=4(a b^(prime)-a^(prime)b)(h a^(prime)-h^(prime)a)