The points at which the function is not precisely defined: x1=1
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0 so we need to solve the equation: −1+log(x)31=0 Solve this equation The points of intersection with the axis X:
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0: substitute x = 0 to 1/(log(x)^3) - 1. −1+log(0)31 The result: f(0)=−1 The point:
(0, -1)
Extrema of the function
In order to find the extrema, we need to solve the equation dxdf(x)=0 (the derivative equals zero), and the roots of this equation are the extrema of this function: dxdf(x)= the first derivative −xlog(x)log(x)33=0 Solve this equation Solutions are not found, function may have no extrema
Inflection points
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 x2log(x)43(1+log(x)4)=0 Solve this equation The roots of this equation x1=e−4 You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function: Points where there is an indetermination: x1=1
x→1−limx2log(x)43(1+log(x)4)=−∞ x→1+limx2log(x)43(1+log(x)4)=∞ - 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−4] Convex at the intervals [e−4,∞)
Vertical asymptotes
Have: x1=1
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo x→−∞lim(−1+log(x)31)=−1 Let's take the limit so, equation of the horizontal asymptote on the left: y=−1 x→∞lim(−1+log(x)31)=−1 Let's take the limit so, equation of the horizontal asymptote on the right: y=−1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 1/(log(x)^3) - 1, divided by x at x->+oo and x ->-oo x→−∞lim(x−1+log(x)31)=0 Let's take the limit so, inclined coincides with the horizontal asymptote on the right x→∞lim(x−1+log(x)31)=0 Let's take the limit so, inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x). So, check: −1+log(x)31=−1+log(−x)31 - No −1+log(x)31=1−log(−x)31 - No so, the function not is neither even, nor odd