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Exercises 30.5 Homework

1.

(Theoretical) For each of the following functions, use the derivative \(f'\) to determine if it is unimodal on the given interval. You do not need to actually find the minimum.

  1. \(f(x) = e^{-x}\sin x \) on \([1,100] \)
  2. \(f(x) = \ln(x)+10/x \) on \([1,100] \)
  3. \(f(x) = x^2 e^{x} \) on \([-10,10] \)
  4. \(f(x) = x^3 e^{x} \) on \([-10,10] \)
2.

One can combine the reliability of golden section method with the speed of Newton's method as follows: start with golden section, and when the bracket becomes small (say, less than \(0.1\)), switch to Newton's method. This makes sense because once we get close to a root of \(f'\text{,}\) Newton's method converges to that specific root very quickly.

Apply the idea of previous paragraph to the minimization problem in Example 30.4.2. This means exiting the while loop sooner, and following Example 30.3.1 after that.