Explain the significance of the length of Delta FF; of the direction of Delta F.

Question

Explain the significance of the length of Delta FF; of the direction of Delta F. Find a point on the graph of F where it is possible that Delta f = Suppose you're to find the maximum of f (x, y, z) what is the significance of the equation Delta f = 0? Find a point on the graph of F where it is possible that Delta f = Suppose you're to find the maximum off (x, y, z) what is the significance of the equation Delta f = 0? Suppose f (x, y) is positive everywhere. Is it possible that Integral Integral_D f dA is negative? Explain. State the second Derivatives Test.

Explanation / Answer

3)
gradF = 0
This will help us in getting the critical points of the function,
i.e the points where the max and min could exist. Because when fx and fy are ZERO, those are the points when the function turns and therefore could represent a max or min
This is the significance.

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4)
f is positive everywhere.
Since the integral represents the area between the curve and x-axis
and because f is positive everuwhere(i.e it lies abobve the x-axis everywhere), the area would ALWAYS BE POSITIVE
So, it is impossible for the integral F.dA to be negative

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5)
Second derivative test :
First we find the criticals using fx = 0, which is the first derivative.
Eqiate that to 0 gives us the criticals.
Now, second derivative test states that :
a) If f''(critical point) < 0, then that critical point is a MAXIMum
B) And if f''(critical) > 0, then that critical point is a MINIMUM

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