Mister Exam
Graphing y = sqrt(x+2*sqrt(x-1))+sqrt(x-2*sqrt(x-1))
The solution
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f(x) = \/ x + 2*\/ x - 1 + \/ x - 2*\/ x - 1
$$f{\left(x \right)} = \sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}}$$
f = sqrt(x - 2*sqrt(x - 1)) + sqrt(x + 2*sqrt(x - 1))
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:
$$\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}} = 0$$
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:
$$\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$\frac{\frac{1}{2} - \frac{1}{2 \sqrt{x - 1}}}{\sqrt{x - 2 \sqrt{x - 1}}} + \frac{\frac{1}{2} + \frac{1}{2 \sqrt{x - 1}}}{\sqrt{x + 2 \sqrt{x - 1}}} = 0$$
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
$$\frac{\frac{1}{2} - \frac{1}{2 \sqrt{x - 1}}}{\sqrt{x - 2 \sqrt{x - 1}}} + \frac{\frac{1}{2} + \frac{1}{2 \sqrt{x - 1}}}{\sqrt{x + 2 \sqrt{x - 1}}} = 0$$
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
$$\frac{- \frac{\left(1 - \frac{1}{\sqrt{x - 1}}\right)^{2}}{\left(x - 2 \sqrt{x - 1}\right)^{\frac{3}{2}}} - \frac{\left(1 + \frac{1}{\sqrt{x - 1}}\right)^{2}}{\left(x + 2 \sqrt{x - 1}\right)^{\frac{3}{2}}} - \frac{1}{\left(x - 1\right)^{\frac{3}{2}} \sqrt{x + 2 \sqrt{x - 1}}} + \frac{1}{\left(x - 1\right)^{\frac{3}{2}} \sqrt{x - 2 \sqrt{x - 1}}}}{4} = 0$$
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
$$\frac{- \frac{\left(1 - \frac{1}{\sqrt{x - 1}}\right)^{2}}{\left(x - 2 \sqrt{x - 1}\right)^{\frac{3}{2}}} - \frac{\left(1 + \frac{1}{\sqrt{x - 1}}\right)^{2}}{\left(x + 2 \sqrt{x - 1}\right)^{\frac{3}{2}}} - \frac{1}{\left(x - 1\right)^{\frac{3}{2}} \sqrt{x + 2 \sqrt{x - 1}}} + \frac{1}{\left(x - 1\right)^{\frac{3}{2}} \sqrt{x - 2 \sqrt{x - 1}}}}{4} = 0$$
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
$$\lim_{x \to -\infty}\left(\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}}\right) = \infty$$
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 + 2*sqrt(x - 1)) + sqrt(x - 2*sqrt(x - 1)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}}}{x}\right) = 0$$
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:
$$\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}} = \sqrt{- x - 2 \sqrt{- x - 1}} + \sqrt{- x + 2 \sqrt{- x - 1}}$$
- No
$$\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}} = - \sqrt{- x - 2 \sqrt{- x - 1}} - \sqrt{- x + 2 \sqrt{- x - 1}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}} = \sqrt{- x - 2 \sqrt{- x - 1}} + \sqrt{- x + 2 \sqrt{- x - 1}}$$
- No
$$\sqrt{x - 2 \sqrt{x - 1}} + \sqrt{x + 2 \sqrt{x - 1}} = - \sqrt{- x - 2 \sqrt{- x - 1}} - \sqrt{- x + 2 \sqrt{- x - 1}}$$
- No
so, the function
not is
neither even, nor odd