Mister Exam
Graphing y = (1-2x)^lnx
The solution
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log(x) f(x) = (1 - 2*x)
$$f{\left(x \right)} = \left(1 - 2 x\right)^{\log{\left(x \right)}}$$
f = (1 - 2*x)^log(x)
The graph of the function
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:
$$\left(1 - 2 x\right)^{\log{\left(x \right)}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\left(1 - 2 x\right)^{\log{\left(x \right)}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\left(1 - 2 x\right)^{\log{\left(x \right)}} \left(- \frac{2 \log{\left(x \right)}}{1 - 2 x} + \frac{\log{\left(1 - 2 x \right)}}{x}\right) = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\left(1 - 2 x\right)^{\log{\left(x \right)}} \left(- \frac{2 \log{\left(x \right)}}{1 - 2 x} + \frac{\log{\left(1 - 2 x \right)}}{x}\right) = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\left(1 - 2 x\right)^{\log{\left(x \right)}} \left(\left(\frac{2 \log{\left(x \right)}}{2 x - 1} + \frac{\log{\left(1 - 2 x \right)}}{x}\right)^{2} - \frac{4 \log{\left(x \right)}}{\left(2 x - 1\right)^{2}} + \frac{4}{x \left(2 x - 1\right)} - \frac{\log{\left(1 - 2 x \right)}}{x^{2}}\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\left(1 - 2 x\right)^{\log{\left(x \right)}} \left(\left(\frac{2 \log{\left(x \right)}}{2 x - 1} + \frac{\log{\left(1 - 2 x \right)}}{x}\right)^{2} - \frac{4 \log{\left(x \right)}}{\left(2 x - 1\right)^{2}} + \frac{4}{x \left(2 x - 1\right)} - \frac{\log{\left(1 - 2 x \right)}}{x^{2}}\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \left(1 - 2 x\right)^{\log{\left(x \right)}} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty} \left(1 - 2 x\right)^{\log{\left(x \right)}} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty} \left(1 - 2 x\right)^{\log{\left(x \right)}} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty} \left(1 - 2 x\right)^{\log{\left(x \right)}} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (1 - 2*x)^log(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(1 - 2 x\right)^{\log{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(1 - 2 x\right)^{\log{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left(1 - 2 x\right)^{\log{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(1 - 2 x\right)^{\log{\left(x \right)}}}{x}\right) = 0$$
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:
$$\left(1 - 2 x\right)^{\log{\left(x \right)}} = \left(2 x + 1\right)^{\log{\left(- x \right)}}$$
- No
$$\left(1 - 2 x\right)^{\log{\left(x \right)}} = - \left(2 x + 1\right)^{\log{\left(- x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(1 - 2 x\right)^{\log{\left(x \right)}} = \left(2 x + 1\right)^{\log{\left(- x \right)}}$$
- No
$$\left(1 - 2 x\right)^{\log{\left(x \right)}} = - \left(2 x + 1\right)^{\log{\left(- x \right)}}$$
- No
so, the function
not is
neither even, nor odd