Mister Exam
Graphing y = -1-2*lambertw(sqrt(exp(x))*exp(-1/2)/2)
The solution
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/ ____ \
| / x -1/2|
|\/ e *e |
f(x) = -1 - 2*W|-------------|
\ 2 /
$$f{\left(x \right)} = - 2 W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right) - 1$$
f = -2*LambertW((exp(-1/2)*sqrt(exp(x)))/2) - 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:
$$- 2 W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right) - 1 = 0$$
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:
$$- 2 W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right) - 1 = 0$$
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
$$- \frac{W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right)}{W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right) + 1} = 0$$
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
$$- \frac{W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right)}{W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right) + 1} = 0$$
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
$$\frac{\left(-1 + \frac{W\left(\frac{e^{\frac{x}{2}}}{2 e^{\frac{1}{2}}}\right)}{W\left(\frac{e^{\frac{x}{2}}}{2 e^{\frac{1}{2}}}\right) + 1}\right) W\left(\frac{e^{\frac{x}{2}}}{2 e^{\frac{1}{2}}}\right)}{2 \left(W\left(\frac{e^{\frac{x}{2}}}{2 e^{\frac{1}{2}}}\right) + 1\right)^{2}} = 0$$
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
$$\frac{\left(-1 + \frac{W\left(\frac{e^{\frac{x}{2}}}{2 e^{\frac{1}{2}}}\right)}{W\left(\frac{e^{\frac{x}{2}}}{2 e^{\frac{1}{2}}}\right) + 1}\right) W\left(\frac{e^{\frac{x}{2}}}{2 e^{\frac{1}{2}}}\right)}{2 \left(W\left(\frac{e^{\frac{x}{2}}}{2 e^{\frac{1}{2}}}\right) + 1\right)^{2}} = 0$$
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
$$\lim_{x \to -\infty}\left(- 2 W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right) - 1\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(- 2 W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right) - 1\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -1$$
$$\lim_{x \to -\infty}\left(- 2 W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right) - 1\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(- 2 W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right) - 1\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -1$$
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:
$$- 2 W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right) - 1 = - 2 W\left(\frac{e^{- \frac{x}{2}}}{2 e^{\frac{1}{2}}}\right) - 1$$
- No
$$- 2 W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right) - 1 = 2 W\left(\frac{e^{- \frac{x}{2}}}{2 e^{\frac{1}{2}}}\right) + 1$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- 2 W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right) - 1 = - 2 W\left(\frac{e^{- \frac{x}{2}}}{2 e^{\frac{1}{2}}}\right) - 1$$
- No
$$- 2 W\left(\frac{e^{- \frac{1}{2}} \sqrt{e^{x}}}{2}\right) - 1 = 2 W\left(\frac{e^{- \frac{x}{2}}}{2 e^{\frac{1}{2}}}\right) + 1$$
- No
so, the function
not is
neither even, nor odd