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200 All boxes with a square base, an open top, and a volume of 50 ft" have a surface area given by S(x) = x" + , where x is the length of the X sides of the base. Find the absolute minimum of the surface area function on the interval (0,co). What are the dimensions of the box with minimum surface area? Determine the derivative of the given function S(x). S'(x) =Determine the intervals on which the following function is concave up or concave down. Identify any inflection points. f(x) = - e"(x- 8) . . . Determine the intervals on which the following functions are concave up or concave down. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.) O A. The function is concave up on and concave down on O B. The function is concave up on O C. The function is concave down onPerform a first derivative test on the function f(x) = - 4x~ + 3x + 2; [ - 5,5]. a. Locate the critical points of the given function. b. Use the first derivative test to locate the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist). . . . a. Locate the critical points of the given function. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The critical point(s) is/are at x = (Simplify your answer. Use a comma to separate answers as needed.) O B. The function does not have a critical point.dT Avalanche forecasters measure the temperature gradient E which is the rate at which the temperature in a snowpack T changes with respect to its depth h. A large temperature gradient may lead to a weak layer in the snowpack. When these weak layers collapse. dT avalanches occur. Avalanche forecasters use the rule ofthumb that if E exceeds 10Cfm anywhere in the snowpack. conditions are iavorable for weaklayer formation. and the risk of avalanche increases. Assume the temperature function is continuous and differentiable. Complete parts (a) through [d] below. a. An avalanche forecaster digs a snow pit and takes two temperature measurements. At the surface (h = D]: the temperature is - 10C. At a depth of 1.? m. the temperature is - 3C. Using the Mean Value Theorem: what can he conclude about the temperature gradient? The average temperature gradient from h = II} to h = 1.? is DC!m. (Simplify your answer. Round to the nearest tenth as needed.) Find the critical points of the following function. 2 f(x) = 5x7- 3x - 1 What is the derivative of f(x) = 5x- - 3x - 1? (x )