Mister Exam

Graphing y = (xln(x)-sqrt(x))/x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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                    ___
       x*log(x) - \/ x 
f(x) = ----------------
              x        
$$f{\left(x \right)} = \frac{- \sqrt{x} + x \log{\left(x \right)}}{x}$$
f = (-sqrt(x) + x*log(x))/x
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
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:
$$\frac{- \sqrt{x} + x \log{\left(x \right)}}{x} = 0$$
Solve this equation
The points of intersection with the axis X:

Analytical solution
$$x_{1} = e^{2 W\left(\frac{1}{2}\right)}$$
Numerical solution
$$x_{1} = 2.02074735861186$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x*log(x) - sqrt(x))/x.
$$\frac{0 \log{\left(0 \right)} - \sqrt{0}}{0}$$
The result:
$$f{\left(0 \right)} = \text{NaN}$$
- the solutions of the equation d'not exist
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
$$\frac{\log{\left(x \right)} + 1 - \frac{1}{2 \sqrt{x}}}{x} - \frac{- \sqrt{x} + x \log{\left(x \right)}}{x^{2}} = 0$$
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
$$\frac{- \frac{2 \log{\left(x \right)} + 2 - \frac{1}{\sqrt{x}}}{x} + \frac{1}{x} - \frac{2 \left(\sqrt{x} - x \log{\left(x \right)}\right)}{x^{2}} + \frac{1}{4 x^{\frac{3}{2}}}}{x} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{- \sqrt{x} + x \log{\left(x \right)}}{x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{- \sqrt{x} + x \log{\left(x \right)}}{x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (x*log(x) - sqrt(x))/x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- \sqrt{x} + x \log{\left(x \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{- \sqrt{x} + x \log{\left(x \right)}}{x^{2}}\right) = 0$$
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:
$$\frac{- \sqrt{x} + x \log{\left(x \right)}}{x} = - \frac{- x \log{\left(- x \right)} - \sqrt{- x}}{x}$$
- No
$$\frac{- \sqrt{x} + x \log{\left(x \right)}}{x} = \frac{- x \log{\left(- x \right)} - \sqrt{- x}}{x}$$
- No
so, the function
not is
neither even, nor odd