Question

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Answer 1

$f(x) = x^2\cdot 2^{-x}\\ \\ f'(x) = 2x\cdot 2^{-x}+x^2\cdot \ln(2)\cdot 2^{-x}\cdot (-1) = \\ \\ = 2^{-x+1}x- \ln(2)x^2\cdot 2^{-x} = 0\\ \\ x\Big(2^{-x+1}-\ln(2)2^{-x}x\Big) = 0 \\ \\ x = 0\quad sau\quad 2^{-x}\cdot 2 = \ln(2) 2^{-x}\cdot x \\ \\ \Rightarrow 2 = \ln(2) \cdot x \Rightarrow x = \dfrac{2}{\ln 2}\\ \\ \Rightarrow \text{Intre radacinile 0 si }\dfrac{2}{\ln 2} \text{ derivata e doar pozitiva sau doar negativa,}\\ \text{iar inafara, semnul e opusul semnului derivatei intre radacini.}\\ \\ \text{Facem x = -1} \Rightarrow f'(-1) = -1\Big(4+2\ln(2)\Big) < 0$

$f'(x) < 0,\quad x \in (-\infty, 0)\cup \Big(\dfrac{2}{\ln 2}, +\infty\Big) \\ f'(x) > 0,\quad x\in \Big(0,\dfrac{2}{\ln 2}\Big) \\ \\ \Rightarrow f(x) \to \text{ strict crescatoare pe }\Big(0,\dfrac{2}{\ln 2}\Big)$