Show (prove) that the following statements are true. (a) Let f be a differentiable function at (a, b) such that it has a horizontal tangent plane at (a, b). Then (a, b) is a critical point of f. (b) Let z = f(z, y) be a differentiable function at (a, 6) such that dz = 0 at (a, b). Then grad f(a, b) = ō'. (c) Let f be a differentiable function at (a, b) such that its directional derivative in the direction of any vector is 0.Then f has a horizontal tangent plane at (a, b).
Show (prove) that the following statements are true. (a) Let f be a differentiable function at (a, b) such that it has a horizontal tangent plane at (a, b). Then (a, b) is a critical point of f. (b) Let z = f(z, y) be a differentiable function at (a, 6) such that dz = 0 at (a, b). Then grad f(a, b) = ō'. (c) Let f be a differentiable function at (a, b) such that its directional derivative in the direction of any vector is 0.Then f has a horizontal tangent plane at (a, b).
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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![Transcription for Educational Website:
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**10. [Omitted Name]. Show (prove) that the following statements are true.**
(a) Let \( f \) be a differentiable function at \( (a, b) \) such that it has a horizontal tangent plane at \( (a, b) \). Then \( (a, b) \) is a critical point of \( f \).
(b) Let \( z = f(x, y) \) be a differentiable function at \( (a, b) \) such that \( dz = 0 \) at \( (a, b) \). Then \( \text{grad } f(a, b) = \mathbf{0} \).
(c) Let \( f \) be a differentiable function at \( (a, b) \) such that its directional derivative in the direction of any vector is 0. Then \( f \) has a horizontal tangent plane at \( (a, b) \).
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This text presents three mathematical propositions concerning differentiable functions and their critical points, gradients, and directional derivatives. Each statement provides a condition under which a particular conclusion is made about the function \( f \) at the point \( (a, b) \).
There are no graphs or diagrams associated with this text.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F10115ea5-5f0e-4fb5-b400-8bf897ed7680%2F426c5178-c7e3-416f-92c0-7ed977f3c08c%2F365hofe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Transcription for Educational Website:
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**10. [Omitted Name]. Show (prove) that the following statements are true.**
(a) Let \( f \) be a differentiable function at \( (a, b) \) such that it has a horizontal tangent plane at \( (a, b) \). Then \( (a, b) \) is a critical point of \( f \).
(b) Let \( z = f(x, y) \) be a differentiable function at \( (a, b) \) such that \( dz = 0 \) at \( (a, b) \). Then \( \text{grad } f(a, b) = \mathbf{0} \).
(c) Let \( f \) be a differentiable function at \( (a, b) \) such that its directional derivative in the direction of any vector is 0. Then \( f \) has a horizontal tangent plane at \( (a, b) \).
---
This text presents three mathematical propositions concerning differentiable functions and their critical points, gradients, and directional derivatives. Each statement provides a condition under which a particular conclusion is made about the function \( f \) at the point \( (a, b) \).
There are no graphs or diagrams associated with this text.
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