For a dosage of x cubic centimeters (cc) of a certain drug, assume that the resu

Question

For a dosage of x cubic centimeters (cc) of a certain drug, assume that the resulting blood pressure B is approximated by B(x) = 0.05x^2 - 0.3x^3. Find the dosage at which the resulting blood pressure is maximized. Round your answer to the nearest hundreth. Show Process

Explanation / Answer

For this question you just need to find out when the B(x) is max/min at interval [0, 0.16]. B(x) = 320² - 1920x³ B'(x) = 640x - 5760x² Let B'(x) = 0 to find the turning point (Local max/min value). Thus 640x - 5760x² = 0 ===> x - 9x² = 0 ===> x(1 - 9x) = 0 ===> x = 0, 1/9, 1/9=0.1111.... within the interval. Determine the feature of these two points by second derivative : B''(x) = 640 - 11520x, B''(0) = 0, ie. x = 0 is a point of inflection. B''(1/9) < 0, x = 1/9 gives a local maximum value. And B(x) is decreasing on (1/9, 0.16]. Thus Bmax = B(1/9) = 320/243 ˜ 1.317 a) The maximum obtained for a dosage of 0.111 cc b) The maximum blood pressure obtained is 1.317

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