Graphing y = xsqrtx-1,5ln_x+2

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           ___   3*log(x)    
f(x) = x*\/ x  - -------- + 2
                    2

f{\left(x \right)} = \left(\sqrt{x} x - \frac{3 \log{\left(x \right)}}{2}\right) + 2

f = sqrt(x)*x - 3*log(x)/2 + 2

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(\sqrt{x} x - \frac{3 \log{\left(x \right)}}{2}\right) + 2 = 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 x*sqrt(x) - 3*log(x)/2 + 2.

\left(- \frac{3 \log{\left(x \right)}}{2} + 0 \sqrt{0}\right) + 2

The result:

f{\left(0 \right)} = 2 - \frac{3 \log{\left(x \right)}}{2}

The point:

(0, 2 - 3*log(x)/2)

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)} =

\frac{3}{4 \sqrt{x}} = 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(\left(\sqrt{x} x - \frac{3 \log{\left(x \right)}}{2}\right) + 2\right) = - \infty i

Let's take the limit
so,
horizontal asymptote on the left doesn’t exist

\lim_{x \to \infty}\left(\left(\sqrt{x} x - \frac{3 \log{\left(x \right)}}{2}\right) + 2\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*sqrt(x) - 3*log(x)/2 + 2, divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{\left(\sqrt{x} x - \frac{3 \log{\left(x \right)}}{2}\right) + 2}{x}\right) = \infty i

Let's take the limit
so,
inclined asymptote on the left doesn’t exist

\lim_{x \to \infty}\left(\frac{\left(\sqrt{x} x - \frac{3 \log{\left(x \right)}}{2}\right) + 2}{x}\right) = \infty

Let's take the limit
so,
inclined asymptote on the right doesn’t exist

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(\sqrt{x} x - \frac{3 \log{\left(x \right)}}{2}\right) + 2 = - x \sqrt{- x} - \frac{3 \log{\left(x \right)}}{2} + 2

- No

\left(\sqrt{x} x - \frac{3 \log{\left(x \right)}}{2}\right) + 2 = x \sqrt{- x} + \frac{3 \log{\left(x \right)}}{2} - 2

- No
so, the function
not is
neither even, nor odd