625-0030-4b65-8a8e-742ad3984061/Assignments/e714ee85-c993-4035-9055-062b...
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DUE DATE: MAY 17,2024
4. OPTIMIZATION
Watch the video 1_Theme 5_Part1 before attempting.
Question 4. \( [\mathbf{2}, \mathbf{9} ; \mathbf{1}, \mathbf{3}, \mathbf{4} ; \mathbf{4}, \mathbf{2} \) marks \( ] \) (a) Consider the following function:
\[
f(x)=\left|x^{2}-2 x-3\right| \text { on }[-4,4]
\]
(i) Prove that \( f(x) \) has a absolute/global maximum and a absolute/global minimum.
(ii) Find the global maximum and minimum.
(b) Consider the following function:
\[
g(x)=x^{6}-1 \text { on }(-1,1)
\]
(i) Explain why the closed interval method of p275 James Stewart cannot be used here (attached with the assignment).
(ii) Prove that \( g(x) \) has a global minimum.
(HINT: Use the open interval test, that is, calculate the limits at the end points) (iii) Find the global minimum.
(c) Consider the following function:
\[
h(x)=(x+2)^{5} \text { on }(-\infty, \infty)
\]
(i) Explain why the closed interval method of p275 James Stewart and the open interval test cannot be used here.
(ii) Prove that \( h(x) \) does not have a maximum or minimum (neither local or global).