`f(x)=e^x-e^(-x)-2sinx-2/3x^3dot` Then the least value of `n` for which `(d^n)/(dx^n)f(x)(""|)_(x=0)i snon z e roi s` (a) 5 (b) 6 (c) 7 (d) 8

The equation of the curve is y=f(x)dot The tangents at [1,f(1),[2,f(2)],a n d[3,f(3)] make angles pi/6,pi/3,a n dpi/4, respectively, with the positive direction of x-axis. Then the value of int_2^3f^(prime)(x)f^(x)dx+int_1^3f^(x)dx is equal to (a) -1/(sqrt(3)) (b) 1/(sqrt(3)) (c) 0 (d) none of these

For x in R , lim_(xrarroo)((x-3)/(x+2))^x is equal to (a) e (b) e^(-1) (c) e^(-5) (d) e^5

Instead of the usual definition of derivative Df(x), if we define a new kind of derivative D^*F(x) by the formula D*f(x)=lim_(h->0)(f^2(x+h)-f^2(x))/h ,w h e r ef^2(x) mean [f(x)]^2 and if f(x)=xlogx ,then D^*f(x)|_(x=e) has the value (A)e (B) 2e (c) 4e (d) none of these

Let f(x)=x^2a n dg(x)=sinxfora l lx in Rdot Then the set of all x satisfying (fogogof)(x)=(gogof)(x),w h e r e(fog)(x)=f(g(x)), is +-sqrt(npi),n in {0,1,2, dot} +-sqrt(npi),n in {1,2, dot} pi/2+2npi,n in { ,-2,-1,0,1,2} 2npi,n in { ,-2,-1,0,1,2, }

The domain of f(x)i s(0,1)dot Then the domain of (f(e^x)+f(1n|x|) is (a) (-1, e) (b) (1, e) (c) (-e ,-1) (d) (-e ,1)

If f(x) =(e^x)/(1+e^x), I_1=int(f(-a))^(f(a)) xg(x(1-x)dx, and I_2=int_(f(-a))^(f(a)) g(x(1-x))dx, then the value of (I_2)/(I_1) is (a) -1 (b) -2 (c) 2 (d) 1

Prove that (d^n)/(dx^n)(e^(2x)+e^(-2x))=2^n[e^(2x)+(-1)^n e^(-2x)]

If intsinx d(secx)=f(x)-g(x)+c ,t h e n f(x)=secx (b) f(x)=tanx g(x)=2x (d) g(x)=x

The n t h derivative of x e^x vanishes when (a) x=0 (b) x=-1 (c) x=-n (d) x=n

The domain of f(x)i s(0,1)dot Then the domain of (f(e^x)+f(1n|x|) is (-1, e) (b) (1, e) (-e ,-1) (d) (-e ,1)