Question

Care este valoarea parametrului m≥0 astfel încât ecuatiile $sin^{4}x+cos^{4}x =m$ si $sin^{6}x+cos^{6}x =m$ sa aiba in R aceleasi solutii?

Answer 1

$\begin{cases}(\sin^2x)^2+(\cos^2 x)^2 = m \\ (\sin^2 x)^3+(\cos^2x)^3 = m\end{cases}\\\\ \\\begin{cases}(\sin^2 x+\cos^2 x)^2-2\sin^2 x\cos^2 x = m \\ (\sin^2 x+\cos^2 x)(\sin^4 x-\sin^2 x\cos^2 x+\cos^4 x) = m\end{cases} \\ \\\\ \begin{cases} 1^2 - \dfrac{(\sin 2x)^2}{2} = m \\ 1\cdot\Bigg[m - \dfrac{(\sin 2x)^2}{4}\Bigg]= m\end{cases}\\ \\\\\text{Notez: }\,\,\sin 2x = t,\quad t\in[-1,1] \\\\\Rightarrow m-\dfrac{t^2}{4}= m \Rightarrow t = 0 \\ \\ \Rightarrow 1-\dfrac{t^2}{4} = m \Rightarrow 1-\dfrac{0}{4} = m\Rightarrow \boxed{m = 1}\,\to\text{ solutie unica}$