Let `g(x)=2f(x/2)+f(2-x)a n df^('')(x)<0AAx in (0,2)dot` Then `g(x)` increases in (a)`(1/2,2)` (b) `(4/3,2)` (c)`(0,2)` (d) `(0,4/3)`

Let g(x)=f(logx)+f(2-logx)a n df^(x)<0AAx in (0,3)dot Then find the interval in which g(x) increases.

Let g(x)=f(x)+f(1-x) and f "(x)>0AAx in (0,1)dot Find the intervals of increase and decrease of g(x)dot

If g(x) =2f(2x^3-3x^2)+f(6x^2-4x^3-3) AA x in R and f''(x) gt 0 AA x in R then g(x) is increasing in the interval

If f(x)=int_(x^2)^(x^2+1)e^-t^2dt , then f(x) increases in (0,2) (b) no value of x (0,oo) (d) (-oo,0)

If f"(x)>0AAx in R ,f'(3)=0,a n df(x)=f("tan"hat2x-2"tan"x+4),0

A function g(x) is defined as g(x)=1/4f(2x^2-1)+1/2f(1-x^2)a n df(x) is an increasing function. Then g(x) is increasing in the interval. (-1,1) (-sqrt(2/3),0)uu(sqrt(2/3),oo) (-sqrt(2/3),sqrt(2/3)) (d) none of these

Given that f'(x)=4x^(3)-3x^(2)+2x-1 find f(x) if f(0)=0 .

Let g(x)=f(x)-1. If f(x)+f(1-x)=2AAx in R , then g(x) is symmetrical about. (a)The origin (b) the line x=1/2 the point (1,0) (d) the point (1/2,0)

Let g(x)=e^(f(x))a n df(x+1)=x+f(x)AAx in Rdot If n in I^+,t h e n(g^(prime)(n+1/2))/(g(n+1/2))-(g^(prime)(1/2))/(g(1/2))= 2(1+1/2+1/3++1/n) 2(1+1/3+1/5+1/(2n-1)) n 1

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