Mister Exam
Graphing y = -sqrt(-x-exp(x))
The solution
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/ x
f(x) = -\/ -x - e
$$f{\left(x \right)} = - \sqrt{- x - e^{x}}$$
f = -sqrt(-x - exp(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:
$$- \sqrt{- x - e^{x}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - W\left(1\right)$$
Numerical solution
$$x_{1} = -0.567143290409784$$
so we need to solve the equation:
$$- \sqrt{- x - e^{x}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - W\left(1\right)$$
Numerical solution
$$x_{1} = -0.567143290409784$$
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
$$- \frac{- \frac{e^{x}}{2} - \frac{1}{2}}{\sqrt{- x - e^{x}}} = 0$$
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
$$- \frac{- \frac{e^{x}}{2} - \frac{1}{2}}{\sqrt{- x - e^{x}}} = 0$$
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
$$\frac{2 e^{x} - \frac{\left(e^{x} + 1\right)^{2}}{x + e^{x}}}{4 \sqrt{- x - e^{x}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0.490073068480548$$
$$x_{2} = 0.490073068480639$$
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:
Have no bends at the whole real axis
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 e^{x} - \frac{\left(e^{x} + 1\right)^{2}}{x + e^{x}}}{4 \sqrt{- x - e^{x}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0.490073068480548$$
$$x_{2} = 0.490073068480639$$
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:
Have no bends at the whole real axis
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- \sqrt{- x - e^{x}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- \sqrt{- x - e^{x}}\right) = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(- \sqrt{- x - e^{x}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- \sqrt{- x - e^{x}}\right) = - \infty i$$
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(-x - exp(x)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(- \frac{\sqrt{- x - e^{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(- \frac{\sqrt{- x - e^{x}}}{x}\right) = - \infty i$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(- \frac{\sqrt{- x - e^{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(- \frac{\sqrt{- x - e^{x}}}{x}\right) = - \infty i$$
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:
$$- \sqrt{- x - e^{x}} = - \sqrt{x - e^{- x}}$$
- No
$$- \sqrt{- x - e^{x}} = \sqrt{x - e^{- x}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- \sqrt{- x - e^{x}} = - \sqrt{x - e^{- x}}$$
- No
$$- \sqrt{- x - e^{x}} = \sqrt{x - e^{- x}}$$
- No
so, the function
not is
neither even, nor odd