`f(x)=x^3a t\ x=0` has inflection point `y ' '=0` at `x=0` and change sign.

f(x)=x^((1)/(5)) at x=0 has inflection point.y'D.N.E. at x=0

The function f(x)=x^((1)/(3))(x-1) has two inflection points has one point of extremum is non-differentiable has range [-3x2^(-(8)/(3)),oo)

The curve f(x)=x^(2/3) ,at the point with x=0 ,has

Concavity and convexity : if f''(x) gt 0 AA x in (a,b) then the curve y=f(x) is concave up ( or convex down) in (a,b) and if f''(x) lt 0 AA x in (a,b) then the curve y=f(x) is concave down (or convex up ) in (a,b) Inflection point : The point where concavity of the curve changes is known as point of inflection (at inflection point f''(x) is equal to 0 or undefined) Exhaustive set of values of 'a' for which the function f(x) =x^(4) +ax^(3)+(3x^(2))/(2)+1 will be concave upward along the entire real line is :

Let (f(x+y)-f(x))/(2)=(f(y)-1)/(2)+xy , for all x,yinR,f(x) is differentiable and f'(0)=1. Let g(x) be a derivable function at x=0 and follows the function rule g((x+y)/(k))=(g(x)+g(y))/(k),kinR,kne0,2andg'(0)-lambdag'(0)ne0. If the graphs of y=f(x) and y=g(x) intersect in coincident points then lambda can take values

Concavity and convexity : if f''(x) gt 0 AA x in (a,b) then the curve y=f(x) is concave up ( or convex down) in (a,b) and if f''(x) lt 0 AA x in (a,b) then the curve y=f(x) is concave down (or convex up ) in (a,b) Inflection point : The point where concavity of the curve changes is known as point of inflection (at inflection point f''(x) is equal to 0 or undefined) Number of point of inflection for f(x) =(x-1)^(3) (x-2)^(2) is :

If f(x)=det[[x,a,aa,x,aa,x,aa,a,x]],=0, then f(x)=0, and f'(x)=0 has common root sum of roots of f(x),=0 is -3a none of these

Consider the function f:(-oo,oo)rarr(-oo,oo) defined by f(x)=(x^2-ax+1)/(x^2+ax+1), 0ltalt2 , and let g(x)=int_0^(e^x) (f\'(t)dt)/(1+t^2) . Which of the following is true? (A) g\'(x) is positive on (-oo,0) and negative on (0,oo) (B) g\'(x) is negative on (-oo,0) and positive on (0,oo) (C) g\'(x) changes sign on both (-oo,0) and (0,oo) (D) g\'(x) does not change sign on (-oo,oo)