Mister Exam
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How to use it?
- Graphing y =:
- (x+2)/(x-1)
- x√2-x
- x^2-x+2
- (x+2)/((x^2+16)×(x^2-1))
- Limit of the function:
-
x*log(2*x+5*x^2)
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Identical expressions
- x*log(two *x+ five *x^ two)
- x multiply by logarithm of (2 multiply by x plus 5 multiply by x squared )
- x multiply by logarithm of (two multiply by x plus five multiply by x to the power of two)
- x*log(2*x+5*x2)
- x*log2*x+5*x2
- x*log(2*x+5*x²)
- x*log(2*x+5*x to the power of 2)
- xlog(2x+5x^2)
- xlog(2x+5x2)
- xlog2x+5x2
- xlog2x+5x^2
-
Similar expressions
- x*log(2*x-5*x^2)
Graphing y = x*log(2*x+5*x^2)
The solution
You have entered
[src]
/ 2\ f(x) = x*log\2*x + 5*x /
$$f{\left(x \right)} = x \log{\left(5 x^{2} + 2 x \right)}$$
f = x*log(5*x^2 + 2*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:
$$x \log{\left(5 x^{2} + 2 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.289897948556636$$
$$x_{2} = -0.689897948556636$$
so we need to solve the equation:
$$x \log{\left(5 x^{2} + 2 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.289897948556636$$
$$x_{2} = -0.689897948556636$$
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
$$\frac{x \left(10 x + 2\right)}{5 x^{2} + 2 x} + \log{\left(5 x^{2} + 2 x \right)} = 0$$
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
$$\frac{x \left(10 x + 2\right)}{5 x^{2} + 2 x} + \log{\left(5 x^{2} + 2 x \right)} = 0$$
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
$$\frac{2 \left(5 - \frac{2 \left(5 x + 1\right)^{2}}{x \left(5 x + 2\right)} + \frac{2 \left(5 x + 1\right)}{x}\right)}{5 x + 2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{2}{5} - \frac{\sqrt{2}}{5}$$
$$x_{2} = - \frac{2}{5} + \frac{\sqrt{2}}{5}$$
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
$$\left[- \frac{2}{5} - \frac{\sqrt{2}}{5}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{2}{5} - \frac{\sqrt{2}}{5}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(5 - \frac{2 \left(5 x + 1\right)^{2}}{x \left(5 x + 2\right)} + \frac{2 \left(5 x + 1\right)}{x}\right)}{5 x + 2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{2}{5} - \frac{\sqrt{2}}{5}$$
$$x_{2} = - \frac{2}{5} + \frac{\sqrt{2}}{5}$$
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
$$\left[- \frac{2}{5} - \frac{\sqrt{2}}{5}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{2}{5} - \frac{\sqrt{2}}{5}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(x \log{\left(5 x^{2} + 2 x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x \log{\left(5 x^{2} + 2 x \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x \log{\left(5 x^{2} + 2 x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x \log{\left(5 x^{2} + 2 x \right)}\right) = \infty$$
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(2*x + 5*x^2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty} \log{\left(5 x^{2} + 2 x \right)} = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \log{\left(5 x^{2} + 2 x \right)} = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \log{\left(5 x^{2} + 2 x \right)} = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \log{\left(5 x^{2} + 2 x \right)} = \infty$$
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:
$$x \log{\left(5 x^{2} + 2 x \right)} = - x \log{\left(5 x^{2} - 2 x \right)}$$
- No
$$x \log{\left(5 x^{2} + 2 x \right)} = x \log{\left(5 x^{2} - 2 x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x \log{\left(5 x^{2} + 2 x \right)} = - x \log{\left(5 x^{2} - 2 x \right)}$$
- No
$$x \log{\left(5 x^{2} + 2 x \right)} = x \log{\left(5 x^{2} - 2 x \right)}$$
- No
so, the function
not is
neither even, nor odd