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: $$\left(\frac{12 x}{25} + \frac{4 x}{5}\right) + \frac{1}{50} = 0$$ Solve this equation The points of intersection with the axis X:
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0: substitute x = 0 to 4*x/5 + 12*x/25 + 1/50. $$\left(\frac{0 \cdot 4}{5} + \frac{0 \cdot 12}{25}\right) + \frac{1}{50}$$ The result: $$f{\left(0 \right)} = \frac{1}{50}$$ The point:
(0, 1/50)
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 $$\frac{32}{25} = 0$$ 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 $$0 = 0$$ 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 $$\lim_{x \to -\infty}\left(\left(\frac{12 x}{25} + \frac{4 x}{5}\right) + \frac{1}{50}\right) = -\infty$$ Let's take the limit so, horizontal asymptote on the left doesn’t exist $$\lim_{x \to \infty}\left(\left(\frac{12 x}{25} + \frac{4 x}{5}\right) + \frac{1}{50}\right) = \infty$$ 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 4*x/5 + 12*x/25 + 1/50, divided by x at x->+oo and x ->-oo $$\lim_{x \to -\infty}\left(\frac{\left(\frac{12 x}{25} + \frac{4 x}{5}\right) + \frac{1}{50}}{x}\right) = \frac{32}{25}$$ Let's take the limit so, inclined asymptote equation on the left: $$y = \frac{32 x}{25}$$ $$\lim_{x \to \infty}\left(\frac{\left(\frac{12 x}{25} + \frac{4 x}{5}\right) + \frac{1}{50}}{x}\right) = \frac{32}{25}$$ Let's take the limit so, inclined asymptote equation on the right: $$y = \frac{32 x}{25}$$
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: $$\left(\frac{12 x}{25} + \frac{4 x}{5}\right) + \frac{1}{50} = \frac{1}{50} - \frac{32 x}{25}$$ - No $$\left(\frac{12 x}{25} + \frac{4 x}{5}\right) + \frac{1}{50} = \frac{32 x}{25} - \frac{1}{50}$$ - No so, the function not is neither even, nor odd
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