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Plotting a function graph/ 3*x*sqrt(5)-2*x

Graphing y = 3*x*sqrt(5)-2*x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 3*x*\/ 5  - 2*x
f(x)=−2x+5⋅3xf{\left(x \right)} = - 2 x + \sqrt{5} \cdot 3 x 
f = -2*x + sqrt(5)*(3*x)
The graph of the function
02468-8-6-4-2-1010-100100
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:
−2x+5⋅3x=0- 2 x + \sqrt{5} \cdot 3 x = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=0x_{1} = 0
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (3*x)*sqrt(5) - 2*x.
0⋅35−00 \cdot 3 \sqrt{5} - 0
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−2+35=0-2 + 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
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
0=00 = 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→−∞(−2x+5⋅3x)=−∞\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→∞(−2x+5⋅3x)=∞\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→−∞(−2x+5⋅3xx)=−2+35\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(−2+35)y = x \left(-2 + 3 \sqrt{5}\right)
lim⁡x→∞(−2x+5⋅3xx)=−2+35\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(−2+35)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:
−2x+5⋅3x=−35x+2x- 2 x + \sqrt{5} \cdot 3 x = - 3 \sqrt{5} x + 2 x
- No
−2x+5⋅3x=−2x+35x- 2 x + \sqrt{5} \cdot 3 x = - 2 x + 3 \sqrt{5} x
- No
so, the function
not is
neither even, nor odd
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