Please solve I will give points and rate
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Please solve I will give points and rate

[From: ] [author: ] [Date: 13-02-16] [Hit: ]
The (vector) gradient ∇f(x, y) points in the direction of greatest increase of f(x, y).(∇f)(x,(∇f)(-2,I dont know what you mean by the direction of f .......
Let f(x,y)=5x^3−4xy^2.

(A) Find the direction of f when f is increasing fastest at point (-2,-3).


(B) What is the rate of increase in that direction?

-
(A)
The (vector) gradient ∇f(x, y) points in the direction of greatest increase of f(x, y).

(∇f)(x,y) = (15x² - 4y²)i + (-8xy)j
(∇f)(-2, -3) = (15(-2)² - 4(-3)²)i + ((-8)(-2)(-3))j
  = 24i - 48j

I don't know what you mean by "the direction of f ". f is a (scalar) function, not a vector; it doesn't have a direction. If you're asking for a unit vector in the direction of greatest increase of f, this is (∇f) / |∇f| .

(B)
Speaking of which, the rate of increase in that direction is |(∇f)(-2, -3)| = √(24² + (-48)²) = √2880 = 24√5 ≈ 53.67.

So that unit vector I mentioned in (A) is (24i - 48j)/(24√5) = (√5/5)i - (2√5/5)j
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