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 derivative(x+1)32log(x+1)−3=0Solve this equationThe roots of this equation
x1=−1+e23You 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((x+1)32log(x+1)−3)=∞x→−1+lim((x+1)32log(x+1)−3)=−∞- 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
[−1+e23,∞)Convex at the intervals
(−∞,−1+e23]