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(2x+1)324(−1+2x+13x−1)=0Solve this equationThe roots of this equation
x1=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=−0.5x→−0.5−lim((2x+1)324(−1+2x+13x−1))=−∞x→−0.5+lim((2x+1)324(−1+2x+13x−1))=−∞- limits are equal, then skip the corresponding 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
[2,∞)Convex at the intervals
(−∞,2]