If fy(Q) and fx(P) are negative, then 0 is larger than these two values, right?? So 0 can't be the least.
Furthermore, saying that fx(S) has more compact level curves that fy(Q) means that it's absolute value is larger. That doesn't mean that fx(S) actually is larger...
The partial derivative indicates the rate of change. For example, a partial derivative of -1 means that I'm going downwards with a rate of 1. A partial derivative of -10 means that I'm going downwards with a rate of 10: this is much faster!
So I expect the level lines with a partial derivative of -10 to lie much closer together than with a partial derivative of -1.
The same way, I expect the level lines with a partial derivative of 10 to lie much closer together than with a partial derivative of 1.
If the partial derivative is 0, then you neither increase neither decrease in that direction.
So, in our case: in which case of fx(S) and fy(Q) are you going downwards the fastest?
In which case of fx(P) and fy(R) are you going upwards the fastest?
If you calculate partial derivatives. Which ones is going downward faster: -10 or -1?
Which one is increasing fastest? +10 or +1? What does that mean for your last two values?
When given a situation like in the picture. Try to give explicit values to the partial derivatives. It makes it easier to see what's going on!
Very Interesting !!!
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