Mister Exam
Graphing y = sqrt(x)^3+4*x^7-2*x
The solution
You have entered
[src]
3
___ 7
f(x) = \/ x + 4*x - 2*x
$$f{\left(x \right)} = - 2 x + \left(\left(\sqrt{x}\right)^{3} + 4 x^{7}\right)$$
f = -2*x + (sqrt(x))^3 + 4*x^7
The graph of the function
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
$$3 \left(56 x^{5} + \frac{1}{4 \sqrt{x}}\right) = 0$$
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
$$3 \left(56 x^{5} + \frac{1}{4 \sqrt{x}}\right) = 0$$
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
$$\lim_{x \to -\infty}\left(- 2 x + \left(\left(\sqrt{x}\right)^{3} + 4 x^{7}\right)\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- 2 x + \left(\left(\sqrt{x}\right)^{3} + 4 x^{7}\right)\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(- 2 x + \left(\left(\sqrt{x}\right)^{3} + 4 x^{7}\right)\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- 2 x + \left(\left(\sqrt{x}\right)^{3} + 4 x^{7}\right)\right) = \infty$$
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 (sqrt(x))^3 + 4*x^7 - 2*x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- 2 x + \left(\left(\sqrt{x}\right)^{3} + 4 x^{7}\right)}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{- 2 x + \left(\left(\sqrt{x}\right)^{3} + 4 x^{7}\right)}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{- 2 x + \left(\left(\sqrt{x}\right)^{3} + 4 x^{7}\right)}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{- 2 x + \left(\left(\sqrt{x}\right)^{3} + 4 x^{7}\right)}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$- 2 x + \left(\left(\sqrt{x}\right)^{3} + 4 x^{7}\right) = - 4 x^{7} + 2 x + \left(- x\right)^{\frac{3}{2}}$$
- No
$$- 2 x + \left(\left(\sqrt{x}\right)^{3} + 4 x^{7}\right) = 4 x^{7} - 2 x - \left(- x\right)^{\frac{3}{2}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- 2 x + \left(\left(\sqrt{x}\right)^{3} + 4 x^{7}\right) = - 4 x^{7} + 2 x + \left(- x\right)^{\frac{3}{2}}$$
- No
$$- 2 x + \left(\left(\sqrt{x}\right)^{3} + 4 x^{7}\right) = 4 x^{7} - 2 x - \left(- x\right)^{\frac{3}{2}}$$
- No
so, the function
not is
neither even, nor odd