Mister Exam
Graphing y = x^log2(x+2)
The solution
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log(x + 2)
----------
log(2)
f(x) = x
$$f{\left(x \right)} = x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}}$$
f = x^(log(x + 2)/log(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:
$$x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
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:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} \left(\frac{\log{\left(x \right)}}{\left(x + 2\right) \log{\left(2 \right)}} + \frac{\log{\left(x + 2 \right)}}{x \log{\left(2 \right)}}\right) = 0$$
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
$$x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} \left(\frac{\log{\left(x \right)}}{\left(x + 2\right) \log{\left(2 \right)}} + \frac{\log{\left(x + 2 \right)}}{x \log{\left(2 \right)}}\right) = 0$$
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
$$\frac{x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} \left(\frac{\left(\frac{\log{\left(x \right)}}{x + 2} + \frac{\log{\left(x + 2 \right)}}{x}\right)^{2}}{\log{\left(2 \right)}} - \frac{\log{\left(x \right)}}{\left(x + 2\right)^{2}} + \frac{2}{x \left(x + 2\right)} - \frac{\log{\left(x + 2 \right)}}{x^{2}}\right)}{\log{\left(2 \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0.198270593872683$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[0.198270593872683, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0.198270593872683\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} \left(\frac{\left(\frac{\log{\left(x \right)}}{x + 2} + \frac{\log{\left(x + 2 \right)}}{x}\right)^{2}}{\log{\left(2 \right)}} - \frac{\log{\left(x \right)}}{\left(x + 2\right)^{2}} + \frac{2}{x \left(x + 2\right)} - \frac{\log{\left(x + 2 \right)}}{x^{2}}\right)}{\log{\left(2 \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0.198270593872683$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[0.198270593872683, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0.198270593872683\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} = \infty$$
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^(log(x + 2)/log(2)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}}}{x}\right) = \infty$$
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:
$$x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} = \left(- x\right)^{\frac{\log{\left(2 - x \right)}}{\log{\left(2 \right)}}}$$
- No
$$x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} = - \left(- x\right)^{\frac{\log{\left(2 - x \right)}}{\log{\left(2 \right)}}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} = \left(- x\right)^{\frac{\log{\left(2 - x \right)}}{\log{\left(2 \right)}}}$$
- No
$$x^{\frac{\log{\left(x + 2 \right)}}{\log{\left(2 \right)}}} = - \left(- x\right)^{\frac{\log{\left(2 - x \right)}}{\log{\left(2 \right)}}}$$
- No
so, the function
not is
neither even, nor odd