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  • How to use it?

  • Graphing y =:
  • -x^2-2x+1
  • 5x^2-3x-1
  • -3x+5
  • 3x^4-4x^3
  • Derivative of:
  • (x^2)^(1/3) (x^2)^(1/3)
  • Integral of d{x}:
  • (x^2)^(1/3) (x^2)^(1/3)
  • Limit of the function:
  • (x^2)^(1/3) (x^2)^(1/3)

  • Identical expressions

  • (x^ two)^(one / three)
  • (x squared ) to the power of (1 divide by 3)
  • (x to the power of two) to the power of (one divide by three)
  • (x2)(1/3)
  • x21/3
  • (x²)^(1/3)
  • (x to the power of 2) to the power of (1/3)
  • x^2^1/3
  • (x^2)^(1 divide by 3)
Plotting a function graph/ (x^2)^(1/3)

Graphing y = (x^2)^(1/3)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          ____
       3 /  2 
f(x) = \/  x
f(x)=x23f{\left(x \right)} = \sqrt[3]{x^{2}} 
f = (x^2)^(1/3)
The graph of the function
02468-8-6-4-2-101005
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:
x23=0\sqrt[3]{x^{2}} = 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 (x^2)^(1/3).
023\sqrt[3]{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
2∣x∣233x=0\frac{2 \left|{x}\right|^{\frac{2}{3}}}{3 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
2(2sign⁡(x)∣x∣3−3∣x∣23x)9x=0\frac{2 \left(\frac{2 \operatorname{sign}{\left(x \right)}}{\sqrt[3]{\left|{x}\right|}} - \frac{3 \left|{x}\right|^{\frac{2}{3}}}{x}\right)}{9 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→−∞x23=∞\lim_{x \to -\infty} \sqrt[3]{x^{2}} = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞x23=∞\lim_{x \to \infty} \sqrt[3]{x^{2}} = \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^2)^(1/3), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(∣x∣23x)=0\lim_{x \to -\infty}\left(\frac{\left|{x}\right|^{\frac{2}{3}}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(∣x∣23x)=0\lim_{x \to \infty}\left(\frac{\left|{x}\right|^{\frac{2}{3}}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
x23=x23\sqrt[3]{x^{2}} = \sqrt[3]{x^{2}}
- Yes
x23=−x23\sqrt[3]{x^{2}} = - \sqrt[3]{x^{2}}
- No
so, the function
is
even
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