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Plotting a function graph/ 6+12x-4xsqrtx

Graphing y = 6+12x-4xsqrtx

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 6 + 12*x - 4*x*\/ x
f(x)=−x4x+(12x+6)f{\left(x \right)} = - \sqrt{x} 4 x + \left(12 x + 6\right) 
f = -sqrt(x)*4*x + 12*x + 6
The graph of the function
2.03.04.05.06.07.08.09.011.010.0-5050
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:
−x4x+(12x+6)=0- \sqrt{x} 4 x + \left(12 x + 6\right) = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=3+123338+33383+3338+33383x_{1} = 3 + \frac{12}{\sqrt[3]{\frac{3 \sqrt{33}}{8} + \frac{333}{8}}} + \sqrt[3]{\frac{3 \sqrt{33}}{8} + \frac{333}{8}}
Numerical solution
x1=9.92923599940955x_{1} = 9.92923599940955
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 6 + 12*x - 4*x*sqrt(x).
−0⋅0⋅4+(0⋅12+6)- \sqrt{0} \cdot 0 \cdot 4 + \left(0 \cdot 12 + 6\right)
The result:
f(0)=6f{\left(0 \right)} = 6
The point:
(0, 6)
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
12−6x=012 - 6 \sqrt{x} = 0
Solve this equation
The roots of this equation
x1=4x_{1} = 4
The values of the extrema at the points:
(4, 22)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
Maxima of the function at points:
x1=4x_{1} = 4
Decreasing at intervals
(−∞,4]\left(-\infty, 4\right]
Increasing at intervals
[4,∞)\left[4, \infty\right)
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
−3x=0- \frac{3}{\sqrt{x}} = 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→−∞(−x4x+(12x+6))=∞i\lim_{x \to -\infty}\left(- \sqrt{x} 4 x + \left(12 x + 6\right)\right) = \infty i
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(−x4x+(12x+6))=−∞\lim_{x \to \infty}\left(- \sqrt{x} 4 x + \left(12 x + 6\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 6 + 12*x - 4*x*sqrt(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−x4x+(12x+6)x)=−∞i\lim_{x \to -\infty}\left(\frac{- \sqrt{x} 4 x + \left(12 x + 6\right)}{x}\right) = - \infty i
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞(−x4x+(12x+6)x)=−∞\lim_{x \to \infty}\left(\frac{- \sqrt{x} 4 x + \left(12 x + 6\right)}{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:
−x4x+(12x+6)=4x−x−12x+6- \sqrt{x} 4 x + \left(12 x + 6\right) = 4 x \sqrt{- x} - 12 x + 6
- No
−x4x+(12x+6)=−4x−x+12x−6- \sqrt{x} 4 x + \left(12 x + 6\right) = - 4 x \sqrt{- x} + 12 x - 6
- No
so, the function
not is
neither even, nor odd
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