Mister Exam
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-
How to use it?
- Graphing y =:
- (x+3)^3/(x+1)^2
-
|x^2-x-2|
- x^2-6x+9
- -x^2+6x-3
- Limit of the function:
-
(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)
Graphing y = (1-4*x)^(1/x)
The solution
You have entered
[src]
x _________ f(x) = \/ 1 - 4*x
$$f{\left(x \right)} = \left(1 - 4 x\right)^{\frac{1}{x}}$$
f = (1 - 4*x)^(1/x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\left(1 - 4 x\right)^{\frac{1}{x}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{4}$$
Numerical solution
$$x_{1} = 0.25$$
so we need to solve the equation:
$$\left(1 - 4 x\right)^{\frac{1}{x}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{4}$$
Numerical solution
$$x_{1} = 0.25$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\left(1 - 4 x\right)^{\frac{1}{x}} \left(- \frac{4}{x \left(1 - 4 x\right)} - \frac{\log{\left(1 - 4 x \right)}}{x^{2}}\right) = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\left(1 - 4 x\right)^{\frac{1}{x}} \left(- \frac{4}{x \left(1 - 4 x\right)} - \frac{\log{\left(1 - 4 x \right)}}{x^{2}}\right) = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\left(1 - 4 x\right)^{\frac{1}{x}} \left(- \frac{16}{\left(4 x - 1\right)^{2}} + \frac{\left(\frac{4}{4 x - 1} - \frac{\log{\left(1 - 4 x \right)}}{x}\right)^{2}}{x} - \frac{8}{x \left(4 x - 1\right)} + \frac{2 \log{\left(1 - 4 x \right)}}{x^{2}}\right)}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -42692.2032023153$$
$$x_{2} = -39481.6943228952$$
$$x_{3} = -54396.8693689511$$
$$x_{4} = -46959.7219038159$$
$$x_{5} = -37336.10461865$$
$$x_{6} = -35185.9025265682$$
$$x_{7} = -31951.022037392$$
$$x_{8} = -53336.5973582993$$
$$x_{9} = -50151.5558163857$$
$$x_{10} = -41623.0307277163$$
$$x_{11} = -33030.6719410248$$
$$x_{12} = -45894.1555212373$$
$$x_{13} = -40552.8736669427$$
$$x_{14} = -26529.4693576037$$
$$x_{15} = -52275.634583075$$
$$x_{16} = -43760.4264017426$$
$$x_{17} = -38409.4523763171$$
$$x_{18} = -25439.861357093$$
$$x_{19} = -36261.6046486508$$
$$x_{20} = -55456.4696083794$$
$$x_{21} = -27617.1751374615$$
$$x_{22} = -49088.3966626226$$
$$x_{23} = -34108.9443783851$$
$$x_{24} = -48024.4602508774$$
$$x_{25} = -56515.4161613263$$
$$x_{26} = -51213.9610747184$$
$$x_{27} = -29787.3087991713$$
$$x_{28} = -44827.7334602$$
$$x_{29} = -28703.0885358965$$
$$x_{30} = -30869.9259675612$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\left(1 - 4 x\right)^{\frac{1}{x}} \left(- \frac{16}{\left(4 x - 1\right)^{2}} + \frac{\left(\frac{4}{4 x - 1} - \frac{\log{\left(1 - 4 x \right)}}{x}\right)^{2}}{x} - \frac{8}{x \left(4 x - 1\right)} + \frac{2 \log{\left(1 - 4 x \right)}}{x^{2}}\right)}{x}\right) = \frac{64}{3 e^{4}}$$
$$\lim_{x \to 0^+}\left(\frac{\left(1 - 4 x\right)^{\frac{1}{x}} \left(- \frac{16}{\left(4 x - 1\right)^{2}} + \frac{\left(\frac{4}{4 x - 1} - \frac{\log{\left(1 - 4 x \right)}}{x}\right)^{2}}{x} - \frac{8}{x \left(4 x - 1\right)} + \frac{2 \log{\left(1 - 4 x \right)}}{x^{2}}\right)}{x}\right) = \frac{64}{3 e^{4}}$$
- 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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\left(1 - 4 x\right)^{\frac{1}{x}} \left(- \frac{16}{\left(4 x - 1\right)^{2}} + \frac{\left(\frac{4}{4 x - 1} - \frac{\log{\left(1 - 4 x \right)}}{x}\right)^{2}}{x} - \frac{8}{x \left(4 x - 1\right)} + \frac{2 \log{\left(1 - 4 x \right)}}{x^{2}}\right)}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -42692.2032023153$$
$$x_{2} = -39481.6943228952$$
$$x_{3} = -54396.8693689511$$
$$x_{4} = -46959.7219038159$$
$$x_{5} = -37336.10461865$$
$$x_{6} = -35185.9025265682$$
$$x_{7} = -31951.022037392$$
$$x_{8} = -53336.5973582993$$
$$x_{9} = -50151.5558163857$$
$$x_{10} = -41623.0307277163$$
$$x_{11} = -33030.6719410248$$
$$x_{12} = -45894.1555212373$$
$$x_{13} = -40552.8736669427$$
$$x_{14} = -26529.4693576037$$
$$x_{15} = -52275.634583075$$
$$x_{16} = -43760.4264017426$$
$$x_{17} = -38409.4523763171$$
$$x_{18} = -25439.861357093$$
$$x_{19} = -36261.6046486508$$
$$x_{20} = -55456.4696083794$$
$$x_{21} = -27617.1751374615$$
$$x_{22} = -49088.3966626226$$
$$x_{23} = -34108.9443783851$$
$$x_{24} = -48024.4602508774$$
$$x_{25} = -56515.4161613263$$
$$x_{26} = -51213.9610747184$$
$$x_{27} = -29787.3087991713$$
$$x_{28} = -44827.7334602$$
$$x_{29} = -28703.0885358965$$
$$x_{30} = -30869.9259675612$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\left(1 - 4 x\right)^{\frac{1}{x}} \left(- \frac{16}{\left(4 x - 1\right)^{2}} + \frac{\left(\frac{4}{4 x - 1} - \frac{\log{\left(1 - 4 x \right)}}{x}\right)^{2}}{x} - \frac{8}{x \left(4 x - 1\right)} + \frac{2 \log{\left(1 - 4 x \right)}}{x^{2}}\right)}{x}\right) = \frac{64}{3 e^{4}}$$
$$\lim_{x \to 0^+}\left(\frac{\left(1 - 4 x\right)^{\frac{1}{x}} \left(- \frac{16}{\left(4 x - 1\right)^{2}} + \frac{\left(\frac{4}{4 x - 1} - \frac{\log{\left(1 - 4 x \right)}}{x}\right)^{2}}{x} - \frac{8}{x \left(4 x - 1\right)} + \frac{2 \log{\left(1 - 4 x \right)}}{x^{2}}\right)}{x}\right) = \frac{64}{3 e^{4}}$$
- 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:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \left(1 - 4 x\right)^{\frac{1}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \left(1 - 4 x\right)^{\frac{1}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty} \left(1 - 4 x\right)^{\frac{1}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \left(1 - 4 x\right)^{\frac{1}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
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
$$\lim_{x \to -\infty}\left(\frac{\left(1 - 4 x\right)^{\frac{1}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(1 - 4 x\right)^{\frac{1}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left(1 - 4 x\right)^{\frac{1}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(1 - 4 x\right)^{\frac{1}{x}}}{x}\right) = 0$$
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:
$$\left(1 - 4 x\right)^{\frac{1}{x}} = \left(4 x + 1\right)^{- \frac{1}{x}}$$
- No
$$\left(1 - 4 x\right)^{\frac{1}{x}} = - \left(4 x + 1\right)^{- \frac{1}{x}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(1 - 4 x\right)^{\frac{1}{x}} = \left(4 x + 1\right)^{- \frac{1}{x}}$$
- No
$$\left(1 - 4 x\right)^{\frac{1}{x}} = - \left(4 x + 1\right)^{- \frac{1}{x}}$$
- No
so, the function
not is
neither even, nor odd