Mister Exam
Graphing y = sqrt(-x^2-12*x+2*sqrt((2*x+3)^3)-9)
The solution
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f(x) = \/ - x - 12*x + 2*\/ (2*x + 3) - 9
$$f{\left(x \right)} = \sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9}$$
f = sqrt(-x^2 - 12*x + 2*sqrt((2*x + 3)^3) - 9)
The graph of the function
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
$$\frac{- x - 6 + \frac{3 \sqrt{\left(2 x + 3\right)^{3}}}{2 x + 3}}{\sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9}} = 0$$
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
$$\frac{- x - 6 + \frac{3 \sqrt{\left(2 x + 3\right)^{3}}}{2 x + 3}}{\sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9}} = 0$$
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
$$- \frac{\frac{\left(x + 6 - \frac{3 \sqrt{\left(2 x + 3\right)^{3}}}{2 x + 3}\right)^{2}}{- x^{2} - 12 x + 2 \sqrt{\left(2 x + 3\right)^{3}} - 9} + 1 - \frac{3 \sqrt{\left(2 x + 3\right)^{3}}}{\left(2 x + 3\right)^{2}}}{\sqrt{- x^{2} - 12 x + 2 \sqrt{\left(2 x + 3\right)^{3}} - 9}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{\frac{\left(x + 6 - \frac{3 \sqrt{\left(2 x + 3\right)^{3}}}{2 x + 3}\right)^{2}}{- x^{2} - 12 x + 2 \sqrt{\left(2 x + 3\right)^{3}} - 9} + 1 - \frac{3 \sqrt{\left(2 x + 3\right)^{3}}}{\left(2 x + 3\right)^{2}}}{\sqrt{- x^{2} - 12 x + 2 \sqrt{\left(2 x + 3\right)^{3}} - 9}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9} = \infty i$$
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 sqrt(-x^2 - 12*x + 2*sqrt((2*x + 3)^3) - 9), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9}}{x}\right) = - i$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - i x$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9}}{x}\right) = i$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = i x$$
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9}}{x}\right) = - i$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - i x$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9}}{x}\right) = i$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = i x$$
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:
$$\sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9} = \sqrt{- x^{2} + 12 x + 2 \sqrt{\left(3 - 2 x\right)^{3}} - 9}$$
- No
$$\sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9} = - \sqrt{- x^{2} + 12 x + 2 \sqrt{\left(3 - 2 x\right)^{3}} - 9}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9} = \sqrt{- x^{2} + 12 x + 2 \sqrt{\left(3 - 2 x\right)^{3}} - 9}$$
- No
$$\sqrt{\left(\left(- x^{2} - 12 x\right) + 2 \sqrt{\left(2 x + 3\right)^{3}}\right) - 9} = - \sqrt{- x^{2} + 12 x + 2 \sqrt{\left(3 - 2 x\right)^{3}} - 9}$$
- No
so, the function
not is
neither even, nor odd