Mister Exam
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-
How to use it?
- Graphing y =:
- x^5-5x^4+5x^3+1
- x^4+8x^2+9
- x-4/3x^3
- -x^4+2x^2
-
Identical expressions
- sqrt(x^ three +6x)
- square root of (x cubed plus 6x)
- square root of (x to the power of three plus 6x)
- √(x^3+6x)
- sqrt(x3+6x)
- sqrtx3+6x
- sqrt(x³+6x)
- sqrt(x to the power of 3+6x)
- sqrtx^3+6x
-
Similar expressions
- sqrt(x^3-6x)
Graphing y = sqrt(x^3+6x)
The solution
You have entered
[src]
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f(x) = \/ x + 6*x
$$f{\left(x \right)} = \sqrt{x^{3} + 6 x}$$
f = sqrt(x^3 + 6*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:
$$\sqrt{x^{3} + 6 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$\sqrt{x^{3} + 6 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
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
$$\frac{\frac{3 x^{2}}{2} + 3}{\sqrt{x^{3} + 6 x}} = 0$$
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
$$\frac{\frac{3 x^{2}}{2} + 3}{\sqrt{x^{3} + 6 x}} = 0$$
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
$$\frac{3 \left(\sqrt{x} - \frac{3 \left(x^{2} + 2\right)^{2}}{4 x^{\frac{3}{2}} \left(x^{2} + 6\right)}\right)}{\sqrt{x^{2} + 6}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \sqrt{-6 + 4 \sqrt{3}}$$
$$x_{2} = \sqrt{-6 + 4 \sqrt{3}}$$
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
$$\left[\sqrt{-6 + 4 \sqrt{3}}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \sqrt{-6 + 4 \sqrt{3}}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{3 \left(\sqrt{x} - \frac{3 \left(x^{2} + 2\right)^{2}}{4 x^{\frac{3}{2}} \left(x^{2} + 6\right)}\right)}{\sqrt{x^{2} + 6}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \sqrt{-6 + 4 \sqrt{3}}$$
$$x_{2} = \sqrt{-6 + 4 \sqrt{3}}$$
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
$$\left[\sqrt{-6 + 4 \sqrt{3}}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \sqrt{-6 + 4 \sqrt{3}}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \sqrt{x^{3} + 6 x} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{x^{3} + 6 x} = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \sqrt{x^{3} + 6 x} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{x^{3} + 6 x} = \infty$$
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^3 + 6*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x^{3} + 6 x}}{x}\right) = - \infty i$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\sqrt{x^{3} + 6 x}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x^{3} + 6 x}}{x}\right) = - \infty i$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\sqrt{x^{3} + 6 x}}{x}\right) = \infty$$
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:
$$\sqrt{x^{3} + 6 x} = \sqrt{- x^{3} - 6 x}$$
- No
$$\sqrt{x^{3} + 6 x} = - \sqrt{- x^{3} - 6 x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{x^{3} + 6 x} = \sqrt{- x^{3} - 6 x}$$
- No
$$\sqrt{x^{3} + 6 x} = - \sqrt{- x^{3} - 6 x}$$
- No
so, the function
not is
neither even, nor odd
The graph