Instructiоns: On а sepаrаte sheet оf paper, answer each оf the exam problems shown below. Write your answers clearly. Unless otherwise stated, you will need to justify your answers to get the full credit. Problem 1. (10 pts) Compute the linear, l(x1,x2){"version":"1.1","math":"(l(x_1,x_2))"}, and quadratic, q(x1,x2){"version":"1.1","math":"( q(x_1,x_2))"}, approximations of the function f(x1,x2)=x13x2+x1x24−2x1+2,{"version":"1.1","math":"f(x_1,x_2)=x_1^3x_2 + x_1x_2^4 - 2x_1+2, "} at the point x(0)=[10]⊤{"version":"1.1","math":"( x^{(0)}=left[begin{array}{cc} 1 & 0 end{array}right]^{top})"}. Problem 2. (10 pts) Given a point x(0)=1{"version":"1.1","math":"( x^{(0)}=1)"} and the equation f(x)=x3+x2−x+1=0,{"version":"1.1","math":"[ f(x)=x^3+x^2 -x +1 = 0,]"} find x(1){"version":"1.1","math":"( x^{(1)})"}using Newton's method of tangents for finding a root of f{"version":"1.1","math":"(f)"}. Problem 3. (10 pts) It is known that the minimizer of a certain unimodal function f(x){"version":"1.1","math":"( f(x))"} is located in the uncertainty interval [−3,8].{"version":"1.1","math":" left[begin{array}{cc} -3,& 8 end{array}right]. "} We plan on using the Fibonacci method to reduce the uncertainty interval. The last useful value of the factor reducing the uncertainty range is 2/3{"version":"1.1","math":"( 2/3 )"}, that is, 1−ρN=F2F3=23{"version":"1.1","math":"1-rho_N=frac{F_2}{F_3}=frac{2}{3} "} What is the minimal number of iterations to box in the minimizer within the range 1.0? Problem 4. (10 pts) Is the quadratic form f=f(x1,x2,x3)=x⊤[−1002−1000−3]x=x⊤Qx,x=[x1x2x3],{"version":"1.1","math":"begin{eqnarray*}f=f(x_1,x_2, x_3)&=&x^{top}left[begin{array}{ccc} -1 & 0 & 0\ 2 & -1 & 0\ 0 & 0 & -3 end{array}right]x\ &=&x^top Q x,quad x=left[begin{array}{c} x_1\ x_2\ x_3 end{array}right],end{eqnarray*}"}positive definite, negative definite, positive semi-definite, negative semi-definite, or indefinite? Justify your answer. Problem 5. (20 pts) (10 pts) Does the function f(x1,x2)=x1x2−x12−2x22−x2+5{"version":"1.1","math":"f(x_1,x_2)=x_1x_2-x_1^2-2x_2^2 - x_2 +5 "} have a minimizer or a maximizer? If it does, then find it; otherwise explain why it does not. (10 pts) Does the function f(x1,x2)=2x12+4x1x2+2x22−x1+3{"version":"1.1","math":"f(x_1,x_2)=2x_1^2+4x_1x_2+2x_2^2 - x_1 +3 "}{"version":"1.1","math":"f(x_1,x_2)=2x_1^2+4x_1x_2+2x_2^2 - x_1 +3 "} have a minimizer or a maximizer? If it does, then find it; otherwise explain why it does not. Problem 6. (15 pts) Bracket the minimizer of f=5x12+x22{"version":"1.1","math":"f=5x_1^2+x_2^2 "} on the line passing through the point x(0)=[−10−5]⊤{"version":"1.1","math":"( x^{(0)}=left[begin{array}{cc} -10 & -5 end{array}right]^{top})"} in the direction d=[10]⊤{"version":"1.1","math":"( d =left[begin{array}{cc} 1 & 0 end{array}right]^{top})"}. Use ε=3{"version":"1.1","math":"( varepsilon = 3)"}. Problem 7. (15 pts) Apply the Newton's algorithm to the function, f(x1,x2)=−3x12+0.5x22−12x1+x2+5.{"version":"1.1","math":" f(x_1,x_2)= -3x_1^2 + 0.5x_2^2 -12x_1+ x_2 + 5. "} The starting point is x(0)=[−10]⊤{"version":"1.1","math":"( x^{(0)}=left[begin{array}{cc} -1 & 0 end{array}right]^{top})"}. Is the point that you obtain a maximizer, a minimizer, or neither? Justify your answer. Problem 8. (10 pts) Minimize f(x1,x2)=x12+x22+x1x2−3x2{"version":"1.1","math":" f(x_1,x_2) = x_1^2 + x_2^2 + x_1 x_2 -3 x_2 "} using the conjugate gradient method. The starting point is x(0)=0{"version":"1.1","math":"( x^{(0)}= 0)"}. *** Congratulations, you are almost done with Midterm Exam 1. DO NOT end the Honorlock session until you have submitted your work to Gradescope. When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF to Gradescope as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to go to Gradescope: Midterm Exam 1 Click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.