Toán Lớp 12: Giải pt: $4^{x^2}=(2x^2-x+1).2^x$
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Toán Lớp 12: Giải pt: $4^{x^2}=(2x^2-x+1).2^x$
Comments ( 1 )
\quad 4^{x^2} = (2x^2 – x + 1).2^x\\
\Leftrightarrow 2^{2x^2} = (2x^2 – x + 1).2^x\\
\Leftrightarrow 2^{2x^2 – x} = 2x^2 – x + 1\\
\text{Đặt}\ t = 2x^2 – x\qquad \left(t \geqslant -\dfrac18\right)\\
\text{Phương trình trở thành:}\\
\quad 2^t = t + 1\\
\text{Áp dụng bất đẳng thức Bernoulli ta được:}\\
\left[\begin{array}{l}2^t \geqslant t + 1 \quad \forall t\in \left[-\dfrac18;0\right]\cup [1;+\infty)\\2^t \leqslant t + 1 \quad \forall t \in [0;1]\end{array}\right.\\
\text{Dấu = xảy ra}\ \Leftrightarrow \left[\begin{array}{l}t = 0\\t = 1\end{array}\right.\\
\Leftrightarrow \left[\begin{array}{l}2x^2 – x = 0\\2x^2 – x = 1\end{array}\right.\\
\Leftrightarrow \left[\begin{array}{l}x = -\dfrac12\\x = 0\\x = \dfrac12\\x = 1\end{array}\right.\\
\text{Vậy}\ S = \left\{-\dfrac12;0;\dfrac12;1\right\}
\end{array}\)