Szczegółowe wyjaśnienie:
a) Asymptotes:
Domain of function [tex]f[/tex]:
[tex]D_f=\mathbb{R}[/tex]
There is no point [tex]x_0[/tex] that does not belong to the domain of the function.
Conclusion: the function has no vertical and oblique asymptotes.
Limits:
[tex]\lim_{x \to -\infty} f(x)=0\\ \lim_{x \to +\infty} f(x)=0[/tex]
Conclusion: [tex]y=0[/tex] is a horizontal bilateral asymptote of a function
b) Extrema:
Derivative of function [tex]f[/tex]:
[tex]\frac{d}{dx}f(x)= \mathrm{exp}(-(2x-3)^2)\cdot(8x-12)[/tex]
[tex]\\\frac{d}{dx}f(x)=0\\\\\mathrm{exp}(-(2x-3)^2)\cdot(8x-12)=0\\\\8x-12=0\\\\x=\frac{3}{2}[/tex]
We research the derivative in the vicinity of this point:
[tex]f'(1)=4e^{-1}\approx1.4715\\\\f'(2)=-4e^{-1}\approx-1.4715[/tex]
Conclusion: there is a local maximum at this point
So:
[tex]f_{max}(\frac{3}{2} )=1[/tex]
c) Inflection:
second order derivative:
[tex]\frac{d^2}{dx^2} f(x)=\mathrm{exp}(-(2x-3)^2)\cdot(8x-12)^2-8\cdot\mathrm{exp}(-(2x-3)^2)[/tex]
[tex]\frac{d^2}{dx^2} f(x)=0\\\\\mathrm{exp}(-(2x-3)^2)\cdot(8x-12)^2-8\cdot\mathrm{exp}(-(2x-3)^2)=0\\\\8\cdot\mathrm{exp}(-(2x-3)^2)\cdot(8x^2-24x+17)=0\\\\8x^2-24x+17=0\\\\\Delta=32\\\\x_1=\frac{\sqrt{2} }{4} +\frac{3}{2} \approx1.8536\\\\x_2=\frac{3}{2} -\frac{\sqrt{2} }{4}\approx1.1464[/tex]
We research the second-order derivative near these points
[tex]x_1[/tex]:
[tex]\\\\f''(\frac{3}{2} )=-8\\\\f''(2)=8\cdot e^{-1}\approx2.943[/tex]
Conclusion: [tex]x_1[/tex] is the point of inflection
function value for [tex]x_1[/tex]
[tex]f(x_1)=f\bigg(\frac{\sqrt{2} }{4}+\frac{3}{2}\bigg)= \mathrm{exp}(-\frac{1}{2} )\approx0.6065[/tex]
[tex]x_2[/tex]:
[tex]f''(1)=8\cdot e^{-1}\approx2.943\\\\f''(\frac{3}{2} )=-8[/tex]
Conclusion: [tex]x_2[/tex] is the point of inflection
function value for [tex]x_2[/tex]
[tex]f(x_2)=f\bigg(\frac{-\sqrt{2} }{4}+\frac{3}{2}\bigg)= \mathrm{exp}(-\frac{1}{2} )\approx0.6065[/tex]
