d) (i Prove that if f has a local maximum or minimum at c, and if f'(c) exists, then f' ( c)= 0. (ii) Find the 1000 th derivative of f(x)=xe-x e) The error function erf ( x ) = injera is used in probability, statistics and engineering. Show that the function y = everf (x) satisfies the differential equation v' = 2xy+ 2/ VIC.lb] {51' Car A is travelling west at 501115 N: and ear B is travelling north at 60 mif. Both are headed for the intersection of the two roads. At What rate are the cars approaching each other when car A is 0.3 mi and car B is 0.4 mi from the intersection? An offshore oil well is located in the ocean at a point W, which is 5 miles from the closest shorepoint A on a straight shoreline. The oil is to be piped to a shorepoint B that is 8 miles from A by piping it on a straight line underwater from W to some shorepoint P between A and B and then on to B via a pipe along the shoreline. If the cost of laying pipe is $100,000 per mile under water and $75,000 per mile over land, where should the point P be located to minimize the cost of laying the pipe