Mathematics - Application of Derivative Question with Solution | TestHub

If $f(x)=\left|\begin{array}{ccc}{\sin }^{2}x& {\cos }^{2}x& \ell \\ {\cos }^{2}x& \ell & {\sin }^{2}x\\ \ell & {\sin }^{2}x& {\cos }^{2}x\end{array}\right|$, where $\ell$ is a root of $16{\ell }^2-16\ell +7=0$, then there exist two real numbers $(a,b)\in \left(0,\frac{\pi }{2}\right)$ and $a<b$ for which $f(x)=0$. Does there exist $c\in (a,b)$ for which $f^{\prime }(c)=0$? Then