Let's find the inflection points, we'll need to solve the equation for this
dx2d2f(x)=0(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
dx2d2f(x)=the second derivativex2log(x)21+log(x)2=0Solve this equationThe roots of this equation
x1=e−2You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=1x→1−lim(x2log(x)21+log(x)2)=−∞x→1+lim(x2log(x)21+log(x)2)=∞- the limits are not equal, so
x1=1- is an inflection point
Сonvexity and concavity intervals:Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,e−2]Convex at the intervals
[e−2,∞)