Question

In: Advanced Math

Consider the function ?: (−1,1) × (−1,1) → ℝ given by ?(?, ?) = sin(?? +...

Consider the function ?: (−1,1) × (−1,1) → ℝ given by ?(?, ?) = sin(?^? + ?? + ?² ).

1. Find a bound for the directional derivative of ? in any direction, i.e. find a constant ? such that |?_??(?, ?)| ≤ ? for all (?, ?) ∈ (−1,1) × (−1,1) and ? ∈ ℝ ² with |?| = 1.

Solutions

Expert Solution

Let the given function be ,

We have to find the maximum value of the directional derivative of the function f(x,y)

For this , we will find gradient of f(x,y) and we know that ,

So ,

Now , since

Therefore , we have the following four cases - i.e., (-1,-1) , (-1,1) , (1,-1) , (1,1), but it has asked to find it in any direction , therefore we will find it at one point (-1,-1)

Therefore ,

i.e.,

Therefore maximum value of directional derivative at this point ,

Hence , the bound of directional derivative , or the maximum value of the directional derivative of the function f(x,y) = 2.997

Colby Messinger answered 1 year ago

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