Q. 2 TF - Calculus

Question

The gradient at a minimum: If a function of three variables, $$f(x, y,z)$$, is differentiable at a point $$(x_{0}, y_{0}, z_{0})$$ where the function has a minimum, what is $$\bigtriangledown f(x_{0}, y_{0}, z_{0})$$ ?

Step-by-Step Solution

Answer

The value of $$\bigtriangledown f(x_{0}, y_{0}, z_{0})$$ is equal to zero.

Step 1. Given Information

A function of three variables, $$f(x, y,z)$$, is differentiable at a point $$(x_{0}, y_{0}, z_{0})$$ where the function has a minimum.

Step 2. Explanation

If $$f(x, y,z)$$, is differentiable at a point $$f(x_{0}, y_{0},z_{0})$$ where the function has a maximum, then $$f(x_{0}, y_{0},z_{0}) \leq f(x,y,z)$$

We know that, the gradient for a local minimum with a horizontal tangent plane at a point called the stationary point is zero.

Hence, $$\bigtriangledown f(x_{0}, y_{0},z_{0})=0$$