Mister Exam
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- The intersection points of functions
-
How to use it?
- Graphing y =:
- (x+3)/(x-4)
- x^3-3x^2+6
- -x^3-3x^2+3
- -x^3+3x+1
-
Identical expressions
- x^ three -6x^ two +13x- eight
- x cubed minus 6x squared plus 13x minus 8
- x to the power of three minus 6x to the power of two plus 13x minus eight
- x3-6x2+13x-8
- x³-6x²+13x-8
- x to the power of 3-6x to the power of 2+13x-8
-
Similar expressions
- x^3+6x^2+13x-8
- x^3-6x^2-13x-8
- x^3-6x^2+13x+8
Graphing y = x^3-6x^2+13x-8
The solution
You have entered
[src]
3 2 f(x) = x - 6*x + 13*x - 8
$$f{\left(x \right)} = x^{3} - 6 x^{2} + 13 x - 8$$
f = x^3 - 6*x^2 + 13*x - 1*8
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:
$$x^{3} - 6 x^{2} + 13 x - 8 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
so we need to solve the equation:
$$x^{3} - 6 x^{2} + 13 x - 8 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
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
$$3 x^{2} - 12 x + 13 = 0$$
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
$$3 x^{2} - 12 x + 13 = 0$$
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
$$6 \left(x - 2\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 2$$
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
$$\left[2, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$6 \left(x - 2\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 2$$
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
$$\left[2, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(x^{3} - 6 x^{2} + 13 x - 8\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x^{3} - 6 x^{2} + 13 x - 8\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x^{3} - 6 x^{2} + 13 x - 8\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x^{3} - 6 x^{2} + 13 x - 8\right) = \infty$$
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 x^3 - 6*x^2 + 13*x - 1*8, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x^{3} - 6 x^{2} + 13 x - 8}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x^{3} - 6 x^{2} + 13 x - 8}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{x^{3} - 6 x^{2} + 13 x - 8}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x^{3} - 6 x^{2} + 13 x - 8}{x}\right) = \infty$$
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:
$$x^{3} - 6 x^{2} + 13 x - 8 = - x^{3} - 6 x^{2} - 13 x - 8$$
- No
$$x^{3} - 6 x^{2} + 13 x - 8 = x^{3} + 6 x^{2} + 13 x + 8$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x^{3} - 6 x^{2} + 13 x - 8 = - x^{3} - 6 x^{2} - 13 x - 8$$
- No
$$x^{3} - 6 x^{2} + 13 x - 8 = x^{3} + 6 x^{2} + 13 x + 8$$
- No
so, the function
not is
neither even, nor odd
The graph