Concept explainers
Given the formula
(a) Determine the maximum and the corresponding value of x for this function analytically (i.e., using differentiation).
(b) Verify that Eq. (13.7) yields the same results based on initial guesses of
(a)
To calculate: The maximum value of the function and the corresponding value of variable
Answer to Problem 1P
Solution:
The maximum value of the function
Explanation of Solution
Given Information:
The function
Calculation:
Evaluate the first order derivative of the function
Equate equation (1) to zero and evaluate critical point.
Therefore, a critical point is at
Evaluate the second order derivative of the function
The sign of the second order derivative at
Substitute
Hence, the maximum value of the function at
(b)
To prove: The results obtained in part (a) using the equation as follows:
using the initial guess of
Explanation of Solution
Given Information:
The function is given as,
The initial guesses are
Formula used:
The expression for the parabolic interpolation is given as,
Here,
Proof:
Rewrite the equation of the parabolic interpolation:
Substitute
Thus, the value of
Substitute
Thus, the value of
Substitute
Thus, the value of
Substitute
Thus, the value of
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Chapter 13 Solutions
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