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Graphing y = sqrt(x)*ln(x-2)

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f(x) = \/ x *log(x - 2)

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

f = sqrt(x)*log(x - 2)

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:

\sqrt{x} \log{\left(x - 2 \right)} = 0

Solve this equation
The points of intersection with the axis X:

Analytical solution

x_{1} = 0

x_{2} = 3

Numerical solution

x_{1} = 3

x_{2} = 0

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(x)*log(x - 2).

\sqrt{0} \log{\left(-2 \right)}

The result:

f{\left(0 \right)} = 0

The point:

(0, 0)

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)} =

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

\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{\sqrt{x}}{\left(x - 2\right)^{2}} + \frac{1}{\sqrt{x} \left(x - 2\right)} - \frac{\log{\left(x - 2 \right)}}{4 x^{\frac{3}{2}}} = 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(\sqrt{x} \log{\left(x - 2 \right)}\right) = \infty i

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

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

\lim_{x \to -\infty}\left(\frac{\log{\left(x - 2 \right)}}{\sqrt{x}}\right) = 0

Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right

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

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

- No

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

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