Mister Exam
Graphing y = arcsin(x-3)/2-(ln(4-x))
The solution
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asin(x - 3)
f(x) = ----------- - log(4 - x)
2
$$f{\left(x \right)} = - \log{\left(4 - x \right)} + \frac{\operatorname{asin}{\left(x - 3 \right)}}{2}$$
f = -log(4 - x) + asin(x - 3)/2
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:
$$- \log{\left(4 - x \right)} + \frac{\operatorname{asin}{\left(x - 3 \right)}}{2} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 3$$
so we need to solve the equation:
$$- \log{\left(4 - x \right)} + \frac{\operatorname{asin}{\left(x - 3 \right)}}{2} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 3$$
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
$$\frac{1}{4 - x} + \frac{1}{2 \sqrt{1 - \left(x - 3\right)^{2}}} = 0$$
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
$$\frac{1}{4 - x} + \frac{1}{2 \sqrt{1 - \left(x - 3\right)^{2}}} = 0$$
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
$$\frac{1}{\left(x - 4\right)^{2}} + \frac{x - 3}{2 \left(1 - \left(x - 3\right)^{2}\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{59}{225 \sqrt[3]{\frac{289}{3375} + \frac{2 \sqrt{321}}{225}}} + \sqrt[3]{\frac{289}{3375} + \frac{2 \sqrt{321}}{225}} + \frac{34}{15}$$
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
$$\left[- \frac{59}{225 \sqrt[3]{\frac{289}{3375} + \frac{2 \sqrt{321}}{225}}} + \sqrt[3]{\frac{289}{3375} + \frac{2 \sqrt{321}}{225}} + \frac{34}{15}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{59}{225 \sqrt[3]{\frac{289}{3375} + \frac{2 \sqrt{321}}{225}}} + \sqrt[3]{\frac{289}{3375} + \frac{2 \sqrt{321}}{225}} + \frac{34}{15}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{1}{\left(x - 4\right)^{2}} + \frac{x - 3}{2 \left(1 - \left(x - 3\right)^{2}\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{59}{225 \sqrt[3]{\frac{289}{3375} + \frac{2 \sqrt{321}}{225}}} + \sqrt[3]{\frac{289}{3375} + \frac{2 \sqrt{321}}{225}} + \frac{34}{15}$$
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
$$\left[- \frac{59}{225 \sqrt[3]{\frac{289}{3375} + \frac{2 \sqrt{321}}{225}}} + \sqrt[3]{\frac{289}{3375} + \frac{2 \sqrt{321}}{225}} + \frac{34}{15}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{59}{225 \sqrt[3]{\frac{289}{3375} + \frac{2 \sqrt{321}}{225}}} + \sqrt[3]{\frac{289}{3375} + \frac{2 \sqrt{321}}{225}} + \frac{34}{15}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
Let's take the limit
so,
equation of the horizontal asymptote on the right:
False
Let's take the limit
so,
equation of the horizontal asymptote on the left:
False
False
Let's take the limit
so,
equation of the horizontal asymptote on the right:
False
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of asin(x - 3)/2 - log(4 - x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{- \log{\left(4 - x \right)} + \frac{\operatorname{asin}{\left(x - 3 \right)}}{2}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{- \log{\left(4 - x \right)} + \frac{\operatorname{asin}{\left(x - 3 \right)}}{2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{- \log{\left(4 - x \right)} + \frac{\operatorname{asin}{\left(x - 3 \right)}}{2}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{- \log{\left(4 - x \right)} + \frac{\operatorname{asin}{\left(x - 3 \right)}}{2}}{x}\right) = 0$$
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:
$$- \log{\left(4 - x \right)} + \frac{\operatorname{asin}{\left(x - 3 \right)}}{2} = - \log{\left(x + 4 \right)} - \frac{\operatorname{asin}{\left(x + 3 \right)}}{2}$$
- No
$$- \log{\left(4 - x \right)} + \frac{\operatorname{asin}{\left(x - 3 \right)}}{2} = \log{\left(x + 4 \right)} + \frac{\operatorname{asin}{\left(x + 3 \right)}}{2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- \log{\left(4 - x \right)} + \frac{\operatorname{asin}{\left(x - 3 \right)}}{2} = - \log{\left(x + 4 \right)} - \frac{\operatorname{asin}{\left(x + 3 \right)}}{2}$$
- No
$$- \log{\left(4 - x \right)} + \frac{\operatorname{asin}{\left(x - 3 \right)}}{2} = \log{\left(x + 4 \right)} + \frac{\operatorname{asin}{\left(x + 3 \right)}}{2}$$
- No
so, the function
not is
neither even, nor odd