[tex]f(x)=x^{2020}-2020 x+1[/tex]
a)
[tex]\int\limits^1_0 {( x^{2020}-2020 x+1+2020x-1)} \, dx =\int\limits^1_0 {( x^{2020})} \, dx =\frac{x^{2021}}{2021} |_0^1=\frac{1}{2021}[/tex]
Vezi tabelul de integrale in atasament
b)
Primitiva functiei f este F, F'(x)=f(x)
Pentru a studia convexitatea trebuie sa calculam derivata de ordin 2
F''(x)=(F'(x))'=f'(x)
[tex]f'(x)=2020x^{2019}-2020=0\\\\[/tex]
[tex]2020x^{2019}-2020=0\\\\\\2020x^{2019}=2020\\\\x^{2019}=1\\\\x=1[/tex]
Tabel semn
x -∞ 1 +∞
F''(x)=f'(x) - - - - - - - -0 + + + + +
F(x) ∩ F(1) ∪
Pe intervalul [1,+∞) functia este convexa
c)
[tex]\int\limits^1_0 ({(-x)^{2020}-2020(-x)+1-x^{2020}+2020x-1})e^x \, dx =\int\limits^1_04040xe^x\ dx[/tex]
Calculam integrala prin integrare prin parti
f=4040x f'=4040
g'=eˣ g=eˣ
[tex]\int\limits^1_0 4040xe^x\ dx=4040xe^x|_0^1-\int\limits^1_0 4040e^x\ dx=4040e-4040e^x|_0^1=4040e-4040e+4040=4040[/tex]
Un alt exercitiu similar de bac il gasesti aici: https://brainly.ro/tema/5783061
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