MATH 203 Final Examination April 2019 Page 3 of 3 [14] 9. Given the function f(:1:) = 21:2 :154. (a) Calculate f' (:17) and use it to determine intervals where the function is increasing, intervals where it is decreasing, and the local extrema (if any). (b) Calculate f\" (:12) and use it to determine intervals where the function is concave upward, intervals where the function is concave downward, and the inection points (if any). (c) Sketch the graph of the function x) using the information obtained above. [5] Bonus Question: Let f be a function which is monotonically decreasing (strictly) and differentiable everywhere on the real axis. Let also 9 = 51:2 + l. Prove that the composite function h = f o g has one and only one critical point, and determine whether it corresponds to a maximum, minimum or inection point of h(:z:). MATH 203 Final Examination December 2022 Page 3 of 3 [13] 11. Given the function f(a:) = x4 8:172. (a) Calculate f'(:v) and use it to determine intervals where the function is increasing, intervals where it is decreasing, and all critical numbers on the xaxis where f (:13) has local maximum or local minimum. (b) Calculate f\" (:15) and use it to determine intervals where the function is concave upward, intervals where the function is concave downward, and the inection points (if any). (0) Sketch the graph of the function f(:v) using the information obtained above. [5] Bonus Question. Let p(a:) = 174 + a2$2 2a2a7 , where a is any real number. Prove that the graph y = p(:1:) has at least one point of local minimum on the interval (1, l) . The present document and the contents thereof are the property and copyright of the professor(s) who prepared this exam at Concordia University. No part of the present document may be used for any purpose other than research or teaching purposes at Concordia University. Furthermore, no part of the present document may be sold, reproduced, republished or redisseminated in any manner or form without the prior written permission of its owner and copyright holder. MATH 203 Final Examination April 2023 Page 3 of 3 [13] 10. Given the function f(x) = 3:173 7:132 + 12x 3. (a) Calculate f, (:17) and use it to determine intervals where the function is increasing, intervals where it is decreasing, and all critical numbers on the :13-axis where f(:1:) has local maximum or local minimum. (b) Calculate f\" (:15) and use it to determine intervals where the function is concave upward, intervals where the function is concave downward, and the inection points (if any). (0) Sketch the graph of the function f (:5) using the information obtained above. [5] Bonus Question. Is it possible to have a function f(:r) such that f(0) = 0, f(2) = 4, and f'(x)