Question

For the function f = x^2y + 2y^2x, at the point P(1,3), what is the direction in which the directional derivative is zero?

(a) \(-13\hat{i} – 24\hat{j} \)

(b) \(13\hat{i} + 24 \hat{j} \)

(c) \(±13\hat{i} ∓24\hat{j} \)

(d) \(∓13\hat{i} ± 24\hat{j} \)

I have been asked this question during an online exam.

This is a very interesting question from Directional Derivative in division Vector Differential Calculus of Engineering Mathematics

Answer 1

Correct choice is (c) \(±13\hat{i} ∓24\hat{j} \)

Explanation: Let the directional derivative is zero along \(a1\hat{i} + a2\hat{j}\)

\(Then ∇ f = \hat{i} \frac{∂f}{∂x} +\hat{j} \frac{∂f}{∂y} = \hat{i} (2xy+2y^2) + \hat{j}(x^2+4xy) \)

\(i.e.[∇ f]_{(1,3)} = 24\hat{i} + 13\hat{j} \)

\([∇ f]_{(1,3)} . (a_1\hat{i} +a_2\hat{j}) = 0 \)

\((a_1\hat{i} +a_2\hat{j}) . (a_1\hat{i} + a_2\hat{j}) = 0 \)

\(24a_1 + 13a_2 = 0          ∴ \frac{a1}{13}=\frac{(-a2)}{24} \)

∴ Direction are \(\overline{u1} = 13\hat{i} – 24\hat{j}, \overline{u2} = -13\hat{i} + 24\hat{j}. \)