The points at which the function is not precisely defined: x1=−1−3.3881317890172⋅10−21i
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: tan(x3−x)1=0 Solve this equation Solution is not found, it's possible that the graph doesn't intersect 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/(sqrt(tan(x^3 - x))). tan(03−0)1 The result: f(0)=∞~ sof doesn't intersect Y
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 −2tan(x3−x)tan(x3−x)(3x2−1)(tan2(x3−x)+1)=0 Solve this equation The roots of this equation x1=−33 x2=33 The values of the extrema at the points:
Intervals of increase and decrease of the function: Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from: Minima of the function at points: x1=−33 The function has no maxima Decreasing at intervals [−33,∞) Increasing at intervals (−∞,−33]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True
Let's take the limit so, equation of the horizontal asymptote on the left: y=x→−∞limtan(x3−x)1
True
Let's take the limit so, equation of the horizontal asymptote on the right: y=x→∞limtan(x3−x)1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 1/(sqrt(tan(x^3 - x))), divided by x at x->+oo and x ->-oo
True
Let's take the limit so, inclined asymptote equation on the left: y=xx→−∞lim(xtan(x3−x)1)
True
Let's take the limit so, inclined asymptote equation on the right: y=xx→∞lim(xtan(x3−x)1)
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: tan(x3−x)1=−tan(x3−x)1 - No tan(x3−x)1=−−tan(x3−x)1 - No so, the function not is neither even, nor odd