Mister Exam
Graphing y = xsqrtx-1,5ln_x+2
The solution
You have entered
[src]
___ 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
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
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
$$\frac{3}{4 \sqrt{x}} = 0$$
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
$$\frac{3}{4 \sqrt{x}} = 0$$
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
$$\lim_{x \to -\infty}\left(\left(\sqrt{x} x - \frac{3 \log{\left(x \right)}}{2}\right) + 2\right) = - \infty i$$
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
$$\lim_{x \to -\infty}\left(\frac{\left(\sqrt{x} x - \frac{3 \log{\left(x \right)}}{2}\right) + 2}{x}\right) = \infty i$$
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
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