Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • x/(x^2-4)
  • (x)^2
  • sqrt(x+1)
  • cbrt((x+6)x^2)

  • Identical expressions

  • cbrt((x+ six)x^ two)
  • cubic root of ((x plus 6)x squared )
  • cubic root of ((x plus six)x to the power of two)
  • cbrt((x+6)x2)
  • cbrtx+6x2
  • cbrt((x+6)x²)
  • cbrt((x+6)x to the power of 2)
  • cbrtx+6x^2

  • Similar expressions

  • cbrt((x-6)x^2)
Plotting a function graph/ cbrt((x+6)x^2)

Graphing y = cbrt((x+6)x^2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          ____________
       3 /          2 
f(x) = \/  (x + 6)*x
f(x)=x2(x+6)3f{\left(x \right)} = \sqrt[3]{x^{2} \left(x + 6\right)} 
f = (x^2*(x + 6))^(1/3)
The graph of the function
02468-8-6-4-2-1010020
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:
x2(x+6)3=0\sqrt[3]{x^{2} \left(x + 6\right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−6x_{1} = -6
x2=0x_{2} = 0
Numerical solution
x1=0x_{1} = 0
x2=−6x_{2} = -6
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to ((x + 6)*x^2)^(1/3).
6⋅023\sqrt[3]{6 \cdot 0^{2}}
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
x+63∣x∣23(x23+2x(x+6)3)x2(x+6)=0\frac{\sqrt[3]{x + 6} \left|{x}\right|^{\frac{2}{3}} \left(\frac{x^{2}}{3} + \frac{2 x \left(x + 6\right)}{3}\right)}{x^{2} \left(x + 6\right)} = 0
Solve this equation
The roots of this equation
x1=−4x_{1} = -4
The values of the extrema at the points:
        2/3 
(-4, 2*2   )


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
(x+4)(2x+63sign⁡(x)∣x∣3+∣x∣23(x+6)23)3(x+6)−(x+4)∣x∣23(x+6)53+2(x+2)∣x∣23x(x+6)23−2(x+4)∣x∣23x(x+6)23x=0\frac{\frac{\left(x + 4\right) \left(\frac{2 \sqrt[3]{x + 6} \operatorname{sign}{\left(x \right)}}{\sqrt[3]{\left|{x}\right|}} + \frac{\left|{x}\right|^{\frac{2}{3}}}{\left(x + 6\right)^{\frac{2}{3}}}\right)}{3 \left(x + 6\right)} - \frac{\left(x + 4\right) \left|{x}\right|^{\frac{2}{3}}}{\left(x + 6\right)^{\frac{5}{3}}} + \frac{2 \left(x + 2\right) \left|{x}\right|^{\frac{2}{3}}}{x \left(x + 6\right)^{\frac{2}{3}}} - \frac{2 \left(x + 4\right) \left|{x}\right|^{\frac{2}{3}}}{x \left(x + 6\right)^{\frac{2}{3}}}}{x} = 0
Solve this equation
The roots of this equation
x1=20926.0478841525x_{1} = 20926.0478841525
x2=40442.1733273205x_{2} = 40442.1733273205
x3=30263.2836749993x_{3} = 30263.2836749993
x4=22624.568071242x_{4} = 22624.568071242
x5=38745.9717821731x_{5} = 38745.9717821731
x6=23473.6471222795x_{6} = 23473.6471222795
x7=42138.2945633354x_{7} = 42138.2945633354
x8=25171.5046699387x_{8} = 25171.5046699387
x9=29414.8026953006x_{9} = 29414.8026953006
x10=35353.2812662006x_{10} = 35353.2812662006
x11=31960.1051794607x_{11} = 31960.1051794607
x12=36201.4940552428x_{12} = 36201.4940552428
x13=37049.6788385142x_{13} = 37049.6788385142
x14=24322.6224640784x_{14} = 24322.6224640784
x15=26020.3029224314x_{15} = 26020.3029224314
x16=39594.0832417035x_{16} = 39594.0832417035
x17=26869.0252348779x_{17} = 26869.0252348779
x18=31111.7165515951x_{18} = 31111.7165515951
x19=19226.9372418016x_{19} = 19226.9372418016
x20=28566.2692968941x_{20} = 28566.2692968941
x21=42986.3280499478x_{21} = 42986.3280499478
x22=41290.2433624204x_{22} = 41290.2433624204
x23=37897.837506221x_{23} = 37897.837506221
x24=33656.7631513868x_{24} = 33656.7631513868
x25=21775.3730730961x_{25} = 21775.3730730961
x26=20076.5758260417x_{26} = 20076.5758260417
x27=27717.678631792x_{27} = 27717.678631792
x28=32808.4530117694x_{28} = 32808.4530117694
x29=34505.0383942055x_{29} = 34505.0383942055
x30=43834.3448550658x_{30} = 43834.3448550658

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
[29414.8026953006,∞)\left[29414.8026953006, \infty\right)
Convex at the intervals
(−∞,19226.9372418016]\left(-\infty, 19226.9372418016\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞x2(x+6)3=∞−13\lim_{x \to -\infty} \sqrt[3]{x^{2} \left(x + 6\right)} = \infty \sqrt[3]{-1}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=∞−13y = \infty \sqrt[3]{-1}
lim⁡x→∞x2(x+6)3=∞\lim_{x \to \infty} \sqrt[3]{x^{2} \left(x + 6\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 + 6)*x^2)^(1/3), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x+63∣x∣23x)=−−13\lim_{x \to -\infty}\left(\frac{\sqrt[3]{x + 6} \left|{x}\right|^{\frac{2}{3}}}{x}\right) = - \sqrt[3]{-1}
Let's take the limit
so,
inclined asymptote equation on the left:
y=−−13xy = - \sqrt[3]{-1} x
lim⁡x→∞(x+63∣x∣23x)=1\lim_{x \to \infty}\left(\frac{\sqrt[3]{x + 6} \left|{x}\right|^{\frac{2}{3}}}{x}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the right:
y=xy = 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:
x2(x+6)3=6−x3∣x∣23\sqrt[3]{x^{2} \left(x + 6\right)} = \sqrt[3]{6 - x} \left|{x}\right|^{\frac{2}{3}}
- No
x2(x+6)3=−6−x3∣x∣23\sqrt[3]{x^{2} \left(x + 6\right)} = - \sqrt[3]{6 - x} \left|{x}\right|^{\frac{2}{3}}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator