5. You are starting a race on the beach at point A to reach a buoy at point B in the ocean, as pictured below (bird's eye View). You can run along the beach twice as fast as you can swim in the ocean. The aim is to nd that the optimal location x to enter the ocean is such that 6 = 30". (a) Let in 2 running speed and v2 2 swimming speed. Show that the total travel time between A and B is (explain your working): TU) = ix+ L'Kbl Jr)3 + bi. V] v; (b) Calculate T'(x). (c) Show that if v1 = 2V2, the optimal angle to enter the ocean is 9 = 30. Hint: Note that TU) has no maximum (you can take arbitrary long detours) and that (blX) sinF} = . Jag. x)2+b 1. Find the equation of the tangent line to f (x) = x atx = 1; 2. For the following function, find the stationary points and use the second derivative test to classify the point(s) as maxima, minima or inflexion points. f(x) =-x'+6x2 -9x +4 3. For the following function and domain, find all global and local minima and maxima h(x) = -x2 +4x +4, for, x E [0, 6] 4. A beam of length 8 m has a deflection y at a distance x from one end given by y= 12 x 103 (x4 - 14x3 + 36x2) . Determine the maximum deflection, and the value of x to achieve this