Item$4$

Item $4$

Develop a Python function to implement parabolic interpolation to locate a minimum. The function should have the following features:

$1.$ Base it on two initial guesses and have the program generate the third initial value at the midpoint of the interval.

$2.$ Check whether the guesses bracket a minimum. If not, the function should not implement the algorithm, but return an error message.

$3.$ Iterate until the relative error falls below a stopping criterion or exceeds a maximum number of iterations.

$4.$ Returns the optimal x and f$($x$),$ the relative error achieved, and the number of iterations taken.

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Use the Python function to solve the following problem.

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f$($x$)=$x$2102$sin$($x$)$fx$=$x$210-2$sinx

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Consider the initial guesses to be $0$ and $4.$

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The minimum of the function occurs at x $=.$

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The number of iterations is $.$