The points at which the function is not precisely defined: x1=1.0471975511966 x2=5.23598775598299
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: 1−2cos(x)sin(x)=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 sin(x)/(1 - 2*cos(x)). 1−2cos(0)sin(0) The result: f(0)=0 The point:
(0, 0)
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 1−2cos(x)cos(x)−(1−2cos(x))22sin2(x)=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 2cos(x)−1−2cos(x)−12(cos(x)+2cos(x)−14sin2(x))+1−2cos(x)−14cos(x)sin(x)=0 Solve this equation The roots of this equation x1=0 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=1.0471975511966 x2=5.23598775598299
x→1.0471975511966−lim2cos(x)−1−2cos(x)−12(cos(x)+2cos(x)−14sin2(x))+1−2cos(x)−14cos(x)sin(x)=4.74636579600418⋅1047 x→1.0471975511966+lim2cos(x)−1−2cos(x)−12(cos(x)+2cos(x)−14sin2(x))+1−2cos(x)−14cos(x)sin(x)=4.74636579600418⋅1047 - limits are equal, then skip the corresponding point x→5.23598775598299−lim2cos(x)−1−2cos(x)−12(cos(x)+2cos(x)−14sin2(x))+1−2cos(x)−14cos(x)sin(x)=4.74636579600418⋅1047 x→5.23598775598299+lim2cos(x)−1−2cos(x)−12(cos(x)+2cos(x)−14sin2(x))+1−2cos(x)−14cos(x)sin(x)=4.74636579600418⋅1047 - 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 (−∞,0] Convex at the intervals [0,∞)
Vertical asymptotes
Have: x1=1.0471975511966 x2=5.23598775598299
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→−∞lim(1−2cos(x)sin(x))
True
Let's take the limit so, equation of the horizontal asymptote on the right: y=x→∞lim(1−2cos(x)sin(x))
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)/(1 - 2*cos(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(x(1−2cos(x))sin(x))
True
Let's take the limit so, inclined asymptote equation on the right: y=xx→∞lim(x(1−2cos(x))sin(x))
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: 1−2cos(x)sin(x)=−1−2cos(x)sin(x) - No 1−2cos(x)sin(x)=1−2cos(x)sin(x) - No so, the function not is neither even, nor odd