5. Higher-order derivatives are derivatives that are taken found after a previous derivative has been taken. For example, we can take the first derivative (y') and the second derivative (y") and the 3rd derivative (y") if we are given the original function (y). Quite often, multiple higher-order derivatives can be found Higher-order derivatives are used in Science (physics) and mathematics (Differential Equations). A differential equation is an equation that contains functions and some higher-order derivatives. Using a formal proof structure (LHS = RHS), prove the differential equation xy' - 3y + y" + 4 = 2x2 given the function y = x3 - 2x2 + 3x. 6. A 5,000 m rectangular area of a field is to be enclosed by a fence, with a moveable inner fence built across the narrow part of the field, as shown. The perimeter fence costs $10/m and the inner fence costs $4/m. Determine the dimensions of the field to minimize the cost. 7. The following table displays the historical number of HIV diagnoses per year in a particular country. Year 1997 1998 1999 2000 2001 2002 2003 2004 2005 Diagnoses 2512 2343 2230 2113 2178 2495 2496 2538 2518 a. Using Desmos, Excel or Curve Expert or another curve modelling program, determine an equation that can be used to model this data b. Using this model, estimate the number of diagnoses in 1996 and in 2006. c. At what rate would the number of diagnoses be changing in 2006? d. Halfway through 2006, the number of new HIV diagnoses was found to be 1232. Assuming this rate stays fairly constant for the remainder of the year, does this new information change the modelling equation? If so, how would this change your answer to part (c)? If you were an advocate for furthering HIV and AIDS research and treatment programs, would you be encouraged or discouraged by these results