Consider the vectors u = (−3/5, −4/5), v = (4/√20, 2/√20), w = (−3/√11, 2/√11). For a functionf (x, y), the directional derivatives of f at some point (a, b) with respect to u and v are given by Duf (a, b) =14/5 and Dvf (a, b) = −22/√20. Determine the value of Dwf (a, b)

The directional derivative of a function f(x, y) at a point (a, b) in the direction of a unit vector w = (w1, w2) is given by the formula:

Dwf(a, b) = w1 * fx(a, b) + w2 * fy(a, b)

where fx(a, b) and fy(a, b) are the partial derivatives of f with respect to x and y, respectively, at the point (a, b).

Given that Duf(a, b) = 14/5 and Dvf(a, b) = -22/√20, we can write the following equations based on the formula for the directional derivative:

-3/5 * fx(a, b) - 4/5 * fy(a, b) = 14/5 4/√20 * fx(a, b) + 2/√20 * fy(a, b) = -22/√20

We can solve this system of equations to find fx(a, b) and fy(a, b).

Multiplying the first equation by 4/√20 and the second equation by -3/5, we get:

-12/5√20 * fx(a, b) - 16/5√20 * fy(a, b) = 56/5√20 -12/5√20 * fx(a, b) - 6/5√20 * fy(a, b) = 66/5√20

Subtracting the second equation from the first, we get:

-10/5√20 * fy(a, b) = -10/5√20

Solving for fy(a, b), we get fy(a, b) = 1.

Substituting fy(a, b) = 1 into the first equation, we get:

-3/5 * fx(a, b) - 4/5 = 14/5

Solving for fx(a, b), we get fx(a, b) = 2.

Finally, we can substitute fx(a, b) = 2 and fy(a, b) = 1 into the formula for the directional derivative to find Dwf(a, b):

Dwf(a, b) = -3/√11 * 2 + 2/√11 * 1 = -6/√11 + 2/√11 = -4/√11.