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  • Graphing y =:
  • x^2+3x+3
  • x^2-3x-4
  • x^2-2x-2
  • (x^2+3)/(x-1) (x^2+3)/(x-1)
  • Limit of the function:
  • (1-4*x)^(1/x) (1-4*x)^(1/x)

  • Identical expressions

  • (one - four *x)^(one /x)
  • (1 minus 4 multiply by x) to the power of (1 divide by x)
  • (one minus four multiply by x) to the power of (one divide by x)
  • (1-4*x)(1/x)
  • 1-4*x1/x
  • (1-4x)^(1/x)
  • (1-4x)(1/x)
  • 1-4x1/x
  • 1-4x^1/x
  • (1-4*x)^(1 divide by x)

  • Similar expressions

  • (1+4*x)^(1/x)
Plotting a function graph/ (1-4*x)^(1/x)

Graphing y = (1-4*x)^(1/x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       x _________
f(x) = \/ 1 - 4*x
f(x)=(1−4x)1xf{\left(x \right)} = \left(1 - 4 x\right)^{\frac{1}{x}} 
f = (1 - 4*x)^(1/x)
The graph of the function
02468-8-6-4-2-10100.01.0
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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:
(1−4x)1x=0\left(1 - 4 x\right)^{\frac{1}{x}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=14x_{1} = \frac{1}{4}
Numerical solution
x1=0.25x_{1} = 0.25
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (1 - 4*x)^(1/x).
(1−0)10\left(1 - 0\right)^{\frac{1}{0}}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
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
(1−4x)1x(−4x(1−4x)−log⁡(1−4x)x2)=0\left(1 - 4 x\right)^{\frac{1}{x}} \left(- \frac{4}{x \left(1 - 4 x\right)} - \frac{\log{\left(1 - 4 x \right)}}{x^{2}}\right) = 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
(1−4x)1x(−16(4x−1)2+(44x−1−log⁡(1−4x)x)2x−8x(4x−1)+2log⁡(1−4x)x2)x=0\frac{\left(1 - 4 x\right)^{\frac{1}{x}} \left(- \frac{16}{\left(4 x - 1\right)^{2}} + \frac{\left(\frac{4}{4 x - 1} - \frac{\log{\left(1 - 4 x \right)}}{x}\right)^{2}}{x} - \frac{8}{x \left(4 x - 1\right)} + \frac{2 \log{\left(1 - 4 x \right)}}{x^{2}}\right)}{x} = 0
Solve this equation
The roots of this equation
x1=−42692.2032023153x_{1} = -42692.2032023153
x2=−39481.6943228952x_{2} = -39481.6943228952
x3=−54396.8693689511x_{3} = -54396.8693689511
x4=−46959.7219038159x_{4} = -46959.7219038159
x5=−37336.10461865x_{5} = -37336.10461865
x6=−35185.9025265682x_{6} = -35185.9025265682
x7=−31951.022037392x_{7} = -31951.022037392
x8=−53336.5973582993x_{8} = -53336.5973582993
x9=−50151.5558163857x_{9} = -50151.5558163857
x10=−41623.0307277163x_{10} = -41623.0307277163
x11=−33030.6719410248x_{11} = -33030.6719410248
x12=−45894.1555212373x_{12} = -45894.1555212373
x13=−40552.8736669427x_{13} = -40552.8736669427
x14=−26529.4693576037x_{14} = -26529.4693576037
x15=−52275.634583075x_{15} = -52275.634583075
x16=−43760.4264017426x_{16} = -43760.4264017426
x17=−38409.4523763171x_{17} = -38409.4523763171
x18=−25439.861357093x_{18} = -25439.861357093
x19=−36261.6046486508x_{19} = -36261.6046486508
x20=−55456.4696083794x_{20} = -55456.4696083794
x21=−27617.1751374615x_{21} = -27617.1751374615
x22=−49088.3966626226x_{22} = -49088.3966626226
x23=−34108.9443783851x_{23} = -34108.9443783851
x24=−48024.4602508774x_{24} = -48024.4602508774
x25=−56515.4161613263x_{25} = -56515.4161613263
x26=−51213.9610747184x_{26} = -51213.9610747184
x27=−29787.3087991713x_{27} = -29787.3087991713
x28=−44827.7334602x_{28} = -44827.7334602
x29=−28703.0885358965x_{29} = -28703.0885358965
x30=−30869.9259675612x_{30} = -30869.9259675612
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=0x_{1} = 0

lim⁡x→0−((1−4x)1x(−16(4x−1)2+(44x−1−log⁡(1−4x)x)2x−8x(4x−1)+2log⁡(1−4x)x2)x)=643e4\lim_{x \to 0^-}\left(\frac{\left(1 - 4 x\right)^{\frac{1}{x}} \left(- \frac{16}{\left(4 x - 1\right)^{2}} + \frac{\left(\frac{4}{4 x - 1} - \frac{\log{\left(1 - 4 x \right)}}{x}\right)^{2}}{x} - \frac{8}{x \left(4 x - 1\right)} + \frac{2 \log{\left(1 - 4 x \right)}}{x^{2}}\right)}{x}\right) = \frac{64}{3 e^{4}}
lim⁡x→0+((1−4x)1x(−16(4x−1)2+(44x−1−log⁡(1−4x)x)2x−8x(4x−1)+2log⁡(1−4x)x2)x)=643e4\lim_{x \to 0^+}\left(\frac{\left(1 - 4 x\right)^{\frac{1}{x}} \left(- \frac{16}{\left(4 x - 1\right)^{2}} + \frac{\left(\frac{4}{4 x - 1} - \frac{\log{\left(1 - 4 x \right)}}{x}\right)^{2}}{x} - \frac{8}{x \left(4 x - 1\right)} + \frac{2 \log{\left(1 - 4 x \right)}}{x^{2}}\right)}{x}\right) = \frac{64}{3 e^{4}}
- limits are equal, then skip the corresponding point

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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(1−4x)1x=1\lim_{x \to -\infty} \left(1 - 4 x\right)^{\frac{1}{x}} = 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=1y = 1
lim⁡x→∞(1−4x)1x=1\lim_{x \to \infty} \left(1 - 4 x\right)^{\frac{1}{x}} = 1
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=1y = 1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (1 - 4*x)^(1/x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((1−4x)1xx)=0\lim_{x \to -\infty}\left(\frac{\left(1 - 4 x\right)^{\frac{1}{x}}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞((1−4x)1xx)=0\lim_{x \to \infty}\left(\frac{\left(1 - 4 x\right)^{\frac{1}{x}}}{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:
(1−4x)1x=(4x+1)−1x\left(1 - 4 x\right)^{\frac{1}{x}} = \left(4 x + 1\right)^{- \frac{1}{x}}
- No
(1−4x)1x=−(4x+1)−1x\left(1 - 4 x\right)^{\frac{1}{x}} = - \left(4 x + 1\right)^{- \frac{1}{x}}
- No
so, the function
not is
neither even, nor odd
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