Mister Exam
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-
How to use it?
- Graphing y =:
- xe^-x
- x^5-x
- x^4-8x^2+9
- x^4+8x^2+9
- Derivative of:
-
3*x-log(x+3)^3
-
Identical expressions
- three *x-log(x+ three)^ three
- 3 multiply by x minus logarithm of (x plus 3) cubed
- three multiply by x minus logarithm of (x plus three) to the power of three
- 3*x-log(x+3)3
- 3*x-logx+33
- 3*x-log(x+3)³
- 3*x-log(x+3) to the power of 3
- 3x-log(x+3)^3
- 3x-log(x+3)3
- 3x-logx+33
- 3x-logx+3^3
-
Similar expressions
- 3*x-log(x-3)^3
- 3*x+log(x+3)^3
Graphing y = 3*x-log(x+3)^3
The solution
You have entered
[src]
3 f(x) = 3*x - log (x + 3)
$$f{\left(x \right)} = 3 x - \log{\left(x + 3 \right)}^{3}$$
f = 3*x - log(x + 3)^3
The graph of the function
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:
$$3 x - \log{\left(x + 3 \right)}^{3} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.787000267629375$$
so we need to solve the equation:
$$3 x - \log{\left(x + 3 \right)}^{3} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.787000267629375$$
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
$$3 - \frac{3 \log{\left(x + 3 \right)}^{2}}{x + 3} = 0$$
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
$$3 - \frac{3 \log{\left(x + 3 \right)}^{2}}{x + 3} = 0$$
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
$$\frac{3 \left(\log{\left(x + 3 \right)} - 2\right) \log{\left(x + 3 \right)}}{\left(x + 3\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -2$$
$$x_{2} = -3 + e^{2}$$
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
$$\left(-\infty, -2\right] \cup \left[-3 + e^{2}, \infty\right)$$
Convex at the intervals
$$\left[-2, -3 + e^{2}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{3 \left(\log{\left(x + 3 \right)} - 2\right) \log{\left(x + 3 \right)}}{\left(x + 3\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -2$$
$$x_{2} = -3 + e^{2}$$
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
$$\left(-\infty, -2\right] \cup \left[-3 + e^{2}, \infty\right)$$
Convex at the intervals
$$\left[-2, -3 + e^{2}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(3 x - \log{\left(x + 3 \right)}^{3}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(3 x - \log{\left(x + 3 \right)}^{3}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(3 x - \log{\left(x + 3 \right)}^{3}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(3 x - \log{\left(x + 3 \right)}^{3}\right) = \infty$$
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 3*x - log(x + 3)^3, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{3 x - \log{\left(x + 3 \right)}^{3}}{x}\right) = 3$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 3 x$$
$$\lim_{x \to \infty}\left(\frac{3 x - \log{\left(x + 3 \right)}^{3}}{x}\right) = 3$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 3 x$$
$$\lim_{x \to -\infty}\left(\frac{3 x - \log{\left(x + 3 \right)}^{3}}{x}\right) = 3$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 3 x$$
$$\lim_{x \to \infty}\left(\frac{3 x - \log{\left(x + 3 \right)}^{3}}{x}\right) = 3$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 3 x$$
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:
$$3 x - \log{\left(x + 3 \right)}^{3} = - 3 x - \log{\left(3 - x \right)}^{3}$$
- No
$$3 x - \log{\left(x + 3 \right)}^{3} = 3 x + \log{\left(3 - x \right)}^{3}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$3 x - \log{\left(x + 3 \right)}^{3} = - 3 x - \log{\left(3 - x \right)}^{3}$$
- No
$$3 x - \log{\left(x + 3 \right)}^{3} = 3 x + \log{\left(3 - x \right)}^{3}$$
- No
so, the function
not is
neither even, nor odd