Will the normalized frequencies corresponding to the local minimums and maximums for the frequency magnitude response $|H(e^{j \omega})|$ of an linear and time-invariant system (with real-valued impulse response $h[n]$) always be exactly the same as the phase angles of the zeroes and poles of the system function $H(z)$ (i.e. Z-transform) of the LTI system?
If they do, please give a sketch of proof. If they don't, please give a counter example.
For example, if a zero (i.e. root) of the system function (i.e. Z-transform of the real-valued impulse response function) is $r_0*e^{j \omega_0}$ ($r_0\neq1$), then will $|H(e^{j \omega_0})|$ be a local minimum for the magnitude response function $|H(e^{j \omega})|$?
Similarly, if a pole of the system function is $r_0*e^{j \omega_0}$ ($r_0\neq1$), then will $|H(e^{j \omega_0})|$ be a local maximum of the magnitude response function?