Automated Audio AGC Amplifier Loop Stability (Electrical & Computer Engineering)

An audio electronics hardware engineer in the UNLV Entertainment Engineering & Design program is tuning an automated gain control (AGC) feedback pre-amplifier circuit for live stage microphone feeds. The small-signal total harmonic distortion energy $E(v, r)$ (in microjoules, $\mu\text{J}$ ) across the feedback loop depends on the input reference bias voltage $v$ (in Volts) and the digital feedback resistance parameter $r$ (in kilo-ohms).

Circuit network analysis models the distortion energy surface over the operating domain by:

E(v, r) = v^2 - 4vr + 5r^2 - 6v + 4

Formulate the first-order partial derivatives $\frac{\partial E}{\partial v}$ and $\frac{\partial E}{\partial r}$ and assemble the gradient vector $\nabla E(v, r)$ .
Set up and solve the simultaneous linear system $\nabla E(v, r) = \mathbf{0}$ to find all quiescent operating bias points $(v_0, r_0)$ .
Compute the second-order partial derivatives $E_{vv}$ , $E_{rr}$ , and the cross-derivative $E_{vr}$ .
Calculate the Hessian discriminant $D = E_{vv}E_{rr} - (E_{vr})^2$ and apply the Second Derivative Test to classify the bias point.
In audio hardware design, determine whether this circuit tuning configuration minimizes signal distortion or risks unstable oscillation. State the minimum distortion energy $E_{\min}$ .