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

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

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The graph:

from to

Intersection points:

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Piecewise:

The solution

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f(x) = -3*x - 3*\/ x
f(x)=−3x−3xf{\left(x \right)} = - 3 \sqrt{x} - 3 x 
f = -3*sqrt(x) - 3*x
The graph of the function
02468-8-6-4-2-10100-50
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:
−3x−3x=0- 3 \sqrt{x} - 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 - 3*sqrt(x).
−0−30- 0 - 3 \sqrt{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
−3−32x=0-3 - \frac{3}{2 \sqrt{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
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
34x32=0\frac{3}{4 x^{\frac{3}{2}}} = 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→−∞(−3x−3x)=∞\lim_{x \to -\infty}\left(- 3 \sqrt{x} - 3 x\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(−3x−3x)=−∞\lim_{x \to \infty}\left(- 3 \sqrt{x} - 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 - 3*sqrt(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−3x−3xx)=−3\lim_{x \to -\infty}\left(\frac{- 3 \sqrt{x} - 3 x}{x}\right) = -3
Let's take the limit
so,
inclined asymptote equation on the left:
y=−3xy = - 3 x
lim⁡x→∞(−3x−3xx)=−3\lim_{x \to \infty}\left(\frac{- 3 \sqrt{x} - 3 x}{x}\right) = -3
Let's take the limit
so,
inclined asymptote equation on the right:
y=−3xy = - 3 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:
−3x−3x=3x−3−x- 3 \sqrt{x} - 3 x = 3 x - 3 \sqrt{- x}
- No
−3x−3x=−3x+3−x- 3 \sqrt{x} - 3 x = - 3 x + 3 \sqrt{- x}
- No
so, the function
not is
neither even, nor odd
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