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Section 14.6: Directional Derivatives & Gradient Vector

Exercise 27

  1. Show that a differentiable function \(f\) decreases most rapidly at x in the direction opposite to the gradient vector, that is, in the direction of \(-\nabla f(\mathbf{x})\).
  2. Use the result of part (a) to find the direction in which the function \(f(x, y) = x^4y - x^2y^3\) decreases fastest at the point \((2, -3)\).