Mister Exam
Graphing y = 3x-ln(x-6)^3
The solution
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3 f(x) = 3*x - log (x - 6)
$$f{\left(x \right)} = 3 x - \log{\left(x - 6 \right)}^{3}$$
f = 3*x - log(x - 6)^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:
$$3 x - \log{\left(x - 6 \right)}^{3} = 0$$
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:
$$3 x - \log{\left(x - 6 \right)}^{3} = 0$$
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
$$3 - \frac{3 \log{\left(x - 6 \right)}^{2}}{x - 6} = 0$$
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
$$3 - \frac{3 \log{\left(x - 6 \right)}^{2}}{x - 6} = 0$$
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
$$\frac{3 \left(\log{\left(x - 6 \right)} - 2\right) \log{\left(x - 6 \right)}}{\left(x - 6\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 7$$
$$x_{2} = 6 + e^{2}$$
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
$$\left(-\infty, 7\right] \cup \left[6 + e^{2}, \infty\right)$$
Convex at the intervals
$$\left[7, 6 + e^{2}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{3 \left(\log{\left(x - 6 \right)} - 2\right) \log{\left(x - 6 \right)}}{\left(x - 6\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 7$$
$$x_{2} = 6 + e^{2}$$
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
$$\left(-\infty, 7\right] \cup \left[6 + e^{2}, \infty\right)$$
Convex at the intervals
$$\left[7, 6 + e^{2}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(3 x - \log{\left(x - 6 \right)}^{3}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(3 x - \log{\left(x - 6 \right)}^{3}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(3 x - \log{\left(x - 6 \right)}^{3}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(3 x - \log{\left(x - 6 \right)}^{3}\right) = \infty$$
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 - 6)^3, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{3 x - \log{\left(x - 6 \right)}^{3}}{x}\right) = 3$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 3 x$$
$$\lim_{x \to \infty}\left(\frac{3 x - \log{\left(x - 6 \right)}^{3}}{x}\right) = 3$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 3 x$$
$$\lim_{x \to -\infty}\left(\frac{3 x - \log{\left(x - 6 \right)}^{3}}{x}\right) = 3$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 3 x$$
$$\lim_{x \to \infty}\left(\frac{3 x - \log{\left(x - 6 \right)}^{3}}{x}\right) = 3$$
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:
$$3 x - \log{\left(x - 6 \right)}^{3} = - 3 x - \log{\left(- x - 6 \right)}^{3}$$
- No
$$3 x - \log{\left(x - 6 \right)}^{3} = 3 x + \log{\left(- x - 6 \right)}^{3}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$3 x - \log{\left(x - 6 \right)}^{3} = - 3 x - \log{\left(- x - 6 \right)}^{3}$$
- No
$$3 x - \log{\left(x - 6 \right)}^{3} = 3 x + \log{\left(- x - 6 \right)}^{3}$$
- No
so, the function
not is
neither even, nor odd