Mister Exam
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-
How to use it?
- Graphing y =:
-
log(log(sqrt(x)))
- e^x
-
y=3x^2+15x+17
-
x^4-(13x^2)+36
-
Identical expressions
- log(log(sqrt(x)))
- logarithm of ( logarithm of ( square root of (x)))
- log(log(√(x)))
- loglogsqrtx
Graphing y = log(log(sqrt(x)))
The solution
You have entered
[src]
/ / ___\\ f(x) = log\log\\/ x //
$$f{\left(x \right)} = \log{\left(\log{\left(\sqrt{x} \right)} \right)}$$
f = log(log(sqrt(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:
$$\log{\left(\log{\left(\sqrt{x} \right)} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = e^{2}$$
Numerical solution
$$x_{1} = 7.38905609893065$$
so we need to solve the equation:
$$\log{\left(\log{\left(\sqrt{x} \right)} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = e^{2}$$
Numerical solution
$$x_{1} = 7.38905609893065$$
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
$$\frac{1}{2 x \log{\left(\sqrt{x} \right)}} = 0$$
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
$$\frac{1}{2 x \log{\left(\sqrt{x} \right)}} = 0$$
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
$$- \frac{2 + \frac{1}{\log{\left(\sqrt{x} \right)}}}{4 x^{2} \log{\left(\sqrt{x} \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = e^{-1}$$
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
$$\left(-\infty, e^{-1}\right]$$
Convex at the intervals
$$\left[e^{-1}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{2 + \frac{1}{\log{\left(\sqrt{x} \right)}}}{4 x^{2} \log{\left(\sqrt{x} \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = e^{-1}$$
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
$$\left(-\infty, e^{-1}\right]$$
Convex at the intervals
$$\left[e^{-1}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \log{\left(\log{\left(\sqrt{x} \right)} \right)} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \log{\left(\log{\left(\sqrt{x} \right)} \right)} = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \log{\left(\log{\left(\sqrt{x} \right)} \right)} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \log{\left(\log{\left(\sqrt{x} \right)} \right)} = \infty$$
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 log(log(sqrt(x))), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\log{\left(\sqrt{x} \right)} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(\log{\left(\sqrt{x} \right)} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\log{\left(\sqrt{x} \right)} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(\log{\left(\sqrt{x} \right)} \right)}}{x}\right) = 0$$
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:
$$\log{\left(\log{\left(\sqrt{x} \right)} \right)} = \log{\left(\log{\left(\sqrt{- x} \right)} \right)}$$
- No
$$\log{\left(\log{\left(\sqrt{x} \right)} \right)} = - \log{\left(\log{\left(\sqrt{- x} \right)} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\log{\left(\log{\left(\sqrt{x} \right)} \right)} = \log{\left(\log{\left(\sqrt{- x} \right)} \right)}$$
- No
$$\log{\left(\log{\left(\sqrt{x} \right)} \right)} = - \log{\left(\log{\left(\sqrt{- x} \right)} \right)}$$
- No
so, the function
not is
neither even, nor odd
The graph