1. Fill in the blank: A curve C has equation g(x, y)=k, where Vg#0 on C. If a differentiable function f(x, y) attains its minimum value on C at a point P, then the vectors V(P) and Vg(P) are necessarily Select one: a. zero b. parallel c. equal d. perpendicular e. nonzerO
1. Fill in the blank: A curve C has equation g(x, y)=k, where Vg#0 on C. If a differentiable function f(x, y) attains its minimum value on C at a point P, then the vectors V(P) and Vg(P) are necessarily Select one: a. zero b. parallel c. equal d. perpendicular e. nonzerO
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:### Problem Statement
**1. Fill in the blank:**
A curve \( C \) has equation \( g(x, y) = k \), where \( \nabla g \neq 0 \) on \( C \). If a differentiable function \( f(x, y) \) attains its minimum value on \( C \) at a point \( P \), then the vectors \( \nabla f(P) \) and \( \nabla g(P) \) are necessarily _________.
**Select one:**
- a. zero
- b. parallel
- c. equal
- d. perpendicular
- e. nonzero
### Explanation
The problem involves a curve defined by an equation and requires understanding the relationship between gradient vectors at a point where a function attains its minimum value. You need to fill in the blank with the correct relationship between the gradient vectors \( \nabla f(P) \) and \( \nabla g(P) \).
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