Mister Exam
Graphing y = 3*x*sqrt(5)-2*x
The solution
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___ f(x) = 3*x*\/ 5 - 2*x
$$f{\left(x \right)} = - 2 x + \sqrt{5} \cdot 3 x$$
f = -2*x + sqrt(5)*(3*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:
$$- 2 x + \sqrt{5} \cdot 3 x = 0$$
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:
$$- 2 x + \sqrt{5} \cdot 3 x = 0$$
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
$$-2 + 3 \sqrt{5} = 0$$
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
$$-2 + 3 \sqrt{5} = 0$$
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
$$0 = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$0 = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- 2 x + \sqrt{5} \cdot 3 x\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- 2 x + \sqrt{5} \cdot 3 x\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(- 2 x + \sqrt{5} \cdot 3 x\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- 2 x + \sqrt{5} \cdot 3 x\right) = \infty$$
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 (3*x)*sqrt(5) - 2*x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- 2 x + \sqrt{5} \cdot 3 x}{x}\right) = -2 + 3 \sqrt{5}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \left(-2 + 3 \sqrt{5}\right)$$
$$\lim_{x \to \infty}\left(\frac{- 2 x + \sqrt{5} \cdot 3 x}{x}\right) = -2 + 3 \sqrt{5}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \left(-2 + 3 \sqrt{5}\right)$$
$$\lim_{x \to -\infty}\left(\frac{- 2 x + \sqrt{5} \cdot 3 x}{x}\right) = -2 + 3 \sqrt{5}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \left(-2 + 3 \sqrt{5}\right)$$
$$\lim_{x \to \infty}\left(\frac{- 2 x + \sqrt{5} \cdot 3 x}{x}\right) = -2 + 3 \sqrt{5}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \left(-2 + 3 \sqrt{5}\right)$$
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:
$$- 2 x + \sqrt{5} \cdot 3 x = - 3 \sqrt{5} x + 2 x$$
- No
$$- 2 x + \sqrt{5} \cdot 3 x = - 2 x + 3 \sqrt{5} x$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- 2 x + \sqrt{5} \cdot 3 x = - 3 \sqrt{5} x + 2 x$$
- No
$$- 2 x + \sqrt{5} \cdot 3 x = - 2 x + 3 \sqrt{5} x$$
- No
so, the function
not is
neither even, nor odd