Mister Exam
Graphing y = -sqrt(-2-exp(4*x)-8*x)/4
The solution
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/ 4*x
-\/ -2 - e - 8*x
f(x) = ----------------------
4
$$f{\left(x \right)} = \frac{\left(-1\right) \sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4}$$
f = (-sqrt(-8*x - exp(4*x) - 2))/4
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:
$$\frac{\left(-1\right) \sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{1}{4} - \frac{W\left(\frac{1}{2 e}\right)}{4}$$
Numerical solution
$$x_{1} = -0.289296237870953$$
so we need to solve the equation:
$$\frac{\left(-1\right) \sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{1}{4} - \frac{W\left(\frac{1}{2 e}\right)}{4}$$
Numerical solution
$$x_{1} = -0.289296237870953$$
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
$$- \frac{- 2 e^{4 x} - 4}{4 \sqrt{- 8 x + \left(- e^{4 x} - 2\right)}} = 0$$
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
$$- \frac{- 2 e^{4 x} - 4}{4 \sqrt{- 8 x + \left(- e^{4 x} - 2\right)}} = 0$$
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
$$- \frac{\frac{\left(e^{4 x} + 2\right)^{2}}{8 x + e^{4 x} + 2} - 2 e^{4 x}}{\sqrt{- 8 x - e^{4 x} - 2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0.0874904347054988$$
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
$$- \frac{\frac{\left(e^{4 x} + 2\right)^{2}}{8 x + e^{4 x} + 2} - 2 e^{4 x}}{\sqrt{- 8 x - e^{4 x} - 2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0.0874904347054988$$
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
$$\lim_{x \to -\infty}\left(\frac{\left(-1\right) \sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(-1\right) \sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4}\right) = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\left(-1\right) \sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(-1\right) \sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4}\right) = - \infty i$$
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(-2 - exp(4*x) - 8*x))/4, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(- \frac{\sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(- \frac{\sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4 x}\right) = - \infty i$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(- \frac{\sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(- \frac{\sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4 x}\right) = - \infty i$$
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:
$$\frac{\left(-1\right) \sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4} = - \frac{\sqrt{8 x - 2 - e^{- 4 x}}}{4}$$
- No
$$\frac{\left(-1\right) \sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4} = \frac{\sqrt{8 x - 2 - e^{- 4 x}}}{4}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\left(-1\right) \sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4} = - \frac{\sqrt{8 x - 2 - e^{- 4 x}}}{4}$$
- No
$$\frac{\left(-1\right) \sqrt{- 8 x + \left(- e^{4 x} - 2\right)}}{4} = \frac{\sqrt{8 x - 2 - e^{- 4 x}}}{4}$$
- No
so, the function
not is
neither even, nor odd