Mister Exam
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How to use it?
- Graphing y =:
- y=2x^3+9x^2+12x
- 4x-2x^2
- -3x-4
- 3x^2+12x+9
- Limit of the function:
-
(1+3*x)^(5/x)
-
Identical expressions
- (one + three *x)^(five /x)
- (1 plus 3 multiply by x) to the power of (5 divide by x)
- (one plus three multiply by x) to the power of (five divide by x)
- (1+3*x)(5/x)
- 1+3*x5/x
- (1+3x)^(5/x)
- (1+3x)(5/x)
- 1+3x5/x
- 1+3x^5/x
- (1+3*x)^(5 divide by x)
-
Similar expressions
- (1-3*x)^(5/x)
Graphing y = (1+3*x)^(5/x)
The solution
You have entered
[src]
5
-
x
f(x) = (1 + 3*x)
$$f{\left(x \right)} = \left(3 x + 1\right)^{\frac{5}{x}}$$
f = (3*x + 1)^(5/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:
$$\left(3 x + 1\right)^{\frac{5}{x}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\left(3 x + 1\right)^{\frac{5}{x}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$\left(3 x + 1\right)^{\frac{5}{x}} \left(\frac{15}{x \left(3 x + 1\right)} - \frac{5 \log{\left(3 x + 1 \right)}}{x^{2}}\right) = 0$$
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
$$\left(3 x + 1\right)^{\frac{5}{x}} \left(\frac{15}{x \left(3 x + 1\right)} - \frac{5 \log{\left(3 x + 1 \right)}}{x^{2}}\right) = 0$$
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
$$\frac{5 \left(3 x + 1\right)^{\frac{5}{x}} \left(- \frac{9}{\left(3 x + 1\right)^{2}} + \frac{5 \left(\frac{3}{3 x + 1} - \frac{\log{\left(3 x + 1 \right)}}{x}\right)^{2}}{x} - \frac{6}{x \left(3 x + 1\right)} + \frac{2 \log{\left(3 x + 1 \right)}}{x^{2}}\right)}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 43431.2848572218$$
$$x_{2} = 48599.3491396833$$
$$x_{3} = 46531.8841755977$$
$$x_{4} = 47565.5863605614$$
$$x_{5} = 44464.7178283447$$
$$x_{6} = 45498.2563494007$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{5 \left(3 x + 1\right)^{\frac{5}{x}} \left(- \frac{9}{\left(3 x + 1\right)^{2}} + \frac{5 \left(\frac{3}{3 x + 1} - \frac{\log{\left(3 x + 1 \right)}}{x}\right)^{2}}{x} - \frac{6}{x \left(3 x + 1\right)} + \frac{2 \log{\left(3 x + 1 \right)}}{x^{2}}\right)}{x}\right) = \frac{2385 e^{15}}{4}$$
$$\lim_{x \to 0^+}\left(\frac{5 \left(3 x + 1\right)^{\frac{5}{x}} \left(- \frac{9}{\left(3 x + 1\right)^{2}} + \frac{5 \left(\frac{3}{3 x + 1} - \frac{\log{\left(3 x + 1 \right)}}{x}\right)^{2}}{x} - \frac{6}{x \left(3 x + 1\right)} + \frac{2 \log{\left(3 x + 1 \right)}}{x^{2}}\right)}{x}\right) = \frac{2385 e^{15}}{4}$$
- 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
$$\frac{5 \left(3 x + 1\right)^{\frac{5}{x}} \left(- \frac{9}{\left(3 x + 1\right)^{2}} + \frac{5 \left(\frac{3}{3 x + 1} - \frac{\log{\left(3 x + 1 \right)}}{x}\right)^{2}}{x} - \frac{6}{x \left(3 x + 1\right)} + \frac{2 \log{\left(3 x + 1 \right)}}{x^{2}}\right)}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 43431.2848572218$$
$$x_{2} = 48599.3491396833$$
$$x_{3} = 46531.8841755977$$
$$x_{4} = 47565.5863605614$$
$$x_{5} = 44464.7178283447$$
$$x_{6} = 45498.2563494007$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{5 \left(3 x + 1\right)^{\frac{5}{x}} \left(- \frac{9}{\left(3 x + 1\right)^{2}} + \frac{5 \left(\frac{3}{3 x + 1} - \frac{\log{\left(3 x + 1 \right)}}{x}\right)^{2}}{x} - \frac{6}{x \left(3 x + 1\right)} + \frac{2 \log{\left(3 x + 1 \right)}}{x^{2}}\right)}{x}\right) = \frac{2385 e^{15}}{4}$$
$$\lim_{x \to 0^+}\left(\frac{5 \left(3 x + 1\right)^{\frac{5}{x}} \left(- \frac{9}{\left(3 x + 1\right)^{2}} + \frac{5 \left(\frac{3}{3 x + 1} - \frac{\log{\left(3 x + 1 \right)}}{x}\right)^{2}}{x} - \frac{6}{x \left(3 x + 1\right)} + \frac{2 \log{\left(3 x + 1 \right)}}{x^{2}}\right)}{x}\right) = \frac{2385 e^{15}}{4}$$
- 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
$$\lim_{x \to -\infty} \left(3 x + 1\right)^{\frac{5}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \left(3 x + 1\right)^{\frac{5}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty} \left(3 x + 1\right)^{\frac{5}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \left(3 x + 1\right)^{\frac{5}{x}} = 1$$
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 + 3*x)^(5/x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(3 x + 1\right)^{\frac{5}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(3 x + 1\right)^{\frac{5}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left(3 x + 1\right)^{\frac{5}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(3 x + 1\right)^{\frac{5}{x}}}{x}\right) = 0$$
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:
$$\left(3 x + 1\right)^{\frac{5}{x}} = \left(1 - 3 x\right)^{- \frac{5}{x}}$$
- No
$$\left(3 x + 1\right)^{\frac{5}{x}} = - \left(1 - 3 x\right)^{- \frac{5}{x}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(3 x + 1\right)^{\frac{5}{x}} = \left(1 - 3 x\right)^{- \frac{5}{x}}$$
- No
$$\left(3 x + 1\right)^{\frac{5}{x}} = - \left(1 - 3 x\right)^{- \frac{5}{x}}$$
- No
so, the function
not is
neither even, nor odd