Mister Exam
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-
How to use it?
- Graphing y =:
- (x-1)/(x+5)
- x³+3x²-9x+1
- x^3+3x^2-24x-21
- x³-2x
- Derivative of:
-
x-(ln(x))^2
-
Identical expressions
- x-(ln(x))^ two
- x minus (ln(x)) squared
- x minus (ln(x)) to the power of two
- x-(ln(x))2
- x-lnx2
- x-(ln(x))²
- x-(ln(x)) to the power of 2
- x-lnx^2
-
Similar expressions
- x+(ln(x))^2
Graphing y = x-(ln(x))^2
The solution
You have entered
[src]
2 f(x) = x - log (x)
$$f{\left(x \right)} = x - \log{\left(x \right)}^{2}$$
f = x - log(x)^2
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:
$$x - \log{\left(x \right)}^{2} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.494866414516531$$
so we need to solve the equation:
$$x - \log{\left(x \right)}^{2} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.494866414516531$$
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
$$1 - \frac{2 \log{\left(x \right)}}{x} = 0$$
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
$$1 - \frac{2 \log{\left(x \right)}}{x} = 0$$
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
$$\frac{2 \left(\log{\left(x \right)} - 1\right)}{x^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = e$$
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
$$\left[e, \infty\right)$$
Convex at the intervals
$$\left(-\infty, e\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(\log{\left(x \right)} - 1\right)}{x^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = e$$
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
$$\left[e, \infty\right)$$
Convex at the intervals
$$\left(-\infty, e\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(x - \log{\left(x \right)}^{2}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x - \log{\left(x \right)}^{2}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x - \log{\left(x \right)}^{2}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x - \log{\left(x \right)}^{2}\right) = \infty$$
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 - log(x)^2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x - \log{\left(x \right)}^{2}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{x - \log{\left(x \right)}^{2}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
$$\lim_{x \to -\infty}\left(\frac{x - \log{\left(x \right)}^{2}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{x - \log{\left(x \right)}^{2}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
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:
$$x - \log{\left(x \right)}^{2} = - x - \log{\left(- x \right)}^{2}$$
- No
$$x - \log{\left(x \right)}^{2} = x + \log{\left(- x \right)}^{2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x - \log{\left(x \right)}^{2} = - x - \log{\left(- x \right)}^{2}$$
- No
$$x - \log{\left(x \right)}^{2} = x + \log{\left(- x \right)}^{2}$$
- No
so, the function
not is
neither even, nor odd