I need help finding the critical point of this group of equations: I\'m not sure

Question

I need help finding the critical point of this group of equations:

I'm not sure if I was supposed to find the values of A, B, and C, but they are:

A = -1

B = 7

C = 3

In case you need the original equations for the entire problem (the system above is the general solution with e raised to an eigenvalue, and its eigenvector):

Xk+1 = .7Xk

Yk+1 = .3Xk + .75Yk

Zk+1 = .25Yk + Zk

Where k is just some constant, and X, Y, Z for k = 0 is [1, 0, 0].

I just have no idea how to find a critical point for a system of 3 equations and 3 variables. Step-by-step help would be appreciated.

Explanation / Answer

critical point is where df/dt = 0 on differentiating we get an extra coeeficient of a for e^ax this changes the equation to

A(0.7)e^0.7t + B(0.75)e^0.75t + Ce^t =0

on solving you get a valuee of t i.e the critical point

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