Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**Objective:** Determine the global extreme values of the function on the given set without using calculus.

**Function:**
\[ f(x, y) = 6e^{-(5x^2 + 9y^2)} \]

**Constraint/Set:**
\[ 5x^2 + 9y^2 \leq 2 \]

**Tasks:**
- Find \( f_{\text{min}} = \)
- Find \( f_{\text{max}} = \)

**Notes:**
- The function \( f(x, y) = 6e^{-(5x^2 + 9y^2)} \) is an exponential function with a Gaussian distribution-like shape, where the argument of the exponent, \(-(5x^2 + 9y^2)\), determines its spread and direction. 
- The constraint \( 5x^2 + 9y^2 \leq 2 \) describes an ellipse in the xy-plane. 

The task is to determine the extreme values by considering the behavior of the function under the constraint of the elliptical form without using traditional calculus methods such as differentiation.
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Transcribed Image Text:**Objective:** Determine the global extreme values of the function on the given set without using calculus. **Function:** \[ f(x, y) = 6e^{-(5x^2 + 9y^2)} \] **Constraint/Set:** \[ 5x^2 + 9y^2 \leq 2 \] **Tasks:** - Find \( f_{\text{min}} = \) - Find \( f_{\text{max}} = \) **Notes:** - The function \( f(x, y) = 6e^{-(5x^2 + 9y^2)} \) is an exponential function with a Gaussian distribution-like shape, where the argument of the exponent, \(-(5x^2 + 9y^2)\), determines its spread and direction. - The constraint \( 5x^2 + 9y^2 \leq 2 \) describes an ellipse in the xy-plane. The task is to determine the extreme values by considering the behavior of the function under the constraint of the elliptical form without using traditional calculus methods such as differentiation.
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