Answer:
Initially, that is, at (1,2), the rate of increase of the function in the direction given = 0
Hence the function is neither increasing nor decreasing initially in the given direction.
Step-by-step explanation:
f(x,y) = 2x² - y²
Rate of Change of the function = ∇f = (fₓ, fᵧ)
And we're told to find the rate of change in a particular direction = ∇f.û
We first obtain the unit vector in that direction, û
Direction = (3,4) - (1,2) = (2,2)
Uni vector = vector/magnitude
Vector = 2î + 2j
magnitude = √(2² + 2²) = √8 = 2 √2
(2î + 2j)/(2√2) = (1/√2)î + (1/√2)j = û
Unit vector in the direction = (1/√2, 1/√2)
f = 2x² - y²
At (1,2)
fₓ = ∂f/∂x = 4x = 4
fᵧ = ∂f/∂y = -2y = -4
∇f = (4,-4)
∇f.û = (4î - 4j).((1/√2)î + (1/√2)j) = (4/√2) - (4/√2) = 0
If it was positive, then the function is increasing, if it was negative, the function is decreasing. Zero means neither of them.
Hence the function is neither increasing nor decreasing initially in the given direction.
