Question

What is a point of inflexion ? Does f(x) = x3

 have a point of inflexion at x = 0?

Answer 1

At point of inflexion, concavity of any function (or curve) changes i.e. the curve becomes convex upwards (or convex : f”(x)>0) to convex downwards (or concave : f”(x)<0) & vice-versa. Simply in a word, it is the point where second derivative of a function is zero.

Now for the curve $f(x)=x^3$

$f'(x)=3x^2$ And. $f''(x)=6x$

And at point x=0 , f”(0)=0

Thus $f(x)=x^3$ has a point of inflexion at x=0 .