Q. 1 TF - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1. Given Information

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

Step 2. Explanation

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

Also, the gradient for a local maximum with a horizontal tangent plane at a point is zero and the point is called the stationary point.

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