Mister Exam
Graphing y = sqrt(-ln(x))
The solution
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_________ f(x) = \/ -log(x)
$$f{\left(x \right)} = \sqrt{- \log{\left(x \right)}}$$
f = sqrt(-log(x))
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:
$$\sqrt{- \log{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
so we need to solve the equation:
$$\sqrt{- \log{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
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
$$\frac{\sqrt{- \log{\left(x \right)}}}{2 x \log{\left(x \right)}} = 0$$
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
$$\frac{\sqrt{- \log{\left(x \right)}}}{2 x \log{\left(x \right)}} = 0$$
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
$$- \frac{\sqrt{- \log{\left(x \right)}} \left(2 + \frac{1}{\log{\left(x \right)}}\right)}{4 x^{2} \log{\left(x \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = e^{- \frac{1}{2}}$$
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
$$\left(-\infty, e^{- \frac{1}{2}}\right]$$
Convex at the intervals
$$\left[e^{- \frac{1}{2}}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{\sqrt{- \log{\left(x \right)}} \left(2 + \frac{1}{\log{\left(x \right)}}\right)}{4 x^{2} \log{\left(x \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = e^{- \frac{1}{2}}$$
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
$$\left(-\infty, e^{- \frac{1}{2}}\right]$$
Convex at the intervals
$$\left[e^{- \frac{1}{2}}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \sqrt{- \log{\left(x \right)}} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{- \log{\left(x \right)}} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \sqrt{- \log{\left(x \right)}} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{- \log{\left(x \right)}} = \infty i$$
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(-log(x)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{- \log{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{- \log{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{- \log{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{- \log{\left(x \right)}}}{x}\right) = 0$$
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:
$$\sqrt{- \log{\left(x \right)}} = \sqrt{- \log{\left(- x \right)}}$$
- No
$$\sqrt{- \log{\left(x \right)}} = - \sqrt{- \log{\left(- x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{- \log{\left(x \right)}} = \sqrt{- \log{\left(- x \right)}}$$
- No
$$\sqrt{- \log{\left(x \right)}} = - \sqrt{- \log{\left(- x \right)}}$$
- No
so, the function
not is
neither even, nor odd