Norm attainment of a class of block operator matrices

Given complex numbers $a,b,c$ and a non-negative continuous function $\phi$ defined on $[0,+\mathrm{\infty })$, consider the $2\times 2$ matrix $$M_t=\left(\begin{array}{cc}a& t\\ ct& b\phi (t)\end{array}\right),t\in [0,+\mathrm{\infty })\mathrm{.}$$ We establish conditions for the strict monotonicity of the norm function $t\mapsto \mathrm{\parallel }{M}_t\mathrm{\parallel }$. As an application, we characterize the norm attainment of the corresponding block operator matrix $$T=\left(\begin{array}{cc}a{I}_H& A\\ c{A}^{\ast }& b\phi (\mathrm{\mid }A\mathrm{\mid })\end{array}\right),$$ where $I_H$ is the identity operator on a Hilbert space $H$ and $A$ is a bounded linear operator from another Hilbert space to $H$.