Finding constant coefficients of a cubic equation using critical points.
View full question and answer details: https://www.wyzant.com/resources/answ... Question: Let f(x )= ax^3 + bx^2 + cx + d. Determine the values of a, b, c and d if f(x) has critical points at x = -1 and x = 2, f(0) = 1 and f ′(0) = 6. Hello, please provide a detailed explanation per step to study. Thank-you! ------------------------ Answered By: Austin B. Rising Undergrad with a Specialization in Calculus I More information: https://www.wyzant.com/Tutors/TX/Fort... ------------------------ Written Explanation: Given:f(x) = ax3 + bx2 + cx + dCritical points: x = -1, 2f(0) = 1f ' (0) = 6c and d can be found quickly. Simply use f(0) to solve for d and f ' (0) to solve for c. (Hint: derive f(x))Critical points are x-values which make a function's derivative either 0 or extend to infinity. We will assume that our critical points test for finite derivatives of f(x). Why? If f ' (-1) and/or f ' (2) extend to infinity, either both a and b are infinitely great, only a is infinitely great, or only b is infinitely great. Thus, there is ambiguity and testing for when f ' (-1) and f ' (2) are equal to 0 is infinitely more plausible.f ' (-1) = 0 AND f ' (2) = 0Using f ' (x) = 3ax2 + 2bx + c,3a - 2b + 6 = 012a + 4b + 6 = 0Setting both equations equal to each other and solving for b yields...b = (-3/2)aSubstituting (-3/2)a into b of either equation (I will use the first) gives us...3a - 2(-3/2)a + 6 = 03a + 3a + 6 = 06a + 6 = 06a = -6a = -1Using either equation to find b yields...3(-1) - 2b + 6 = 06 - 3 - 2b = 03 - 2b = 02b = 3b = 3/2Thus, a, b, c, and d are found...a = -1b = 3/2c = 6d = 1Our complete f(x) is...f(x) = -x3 + (3/2)x2 + 6x + 1 See full answer: https://www.wyzant.com/resources/answ... ------------------------ About: Wyzant Ask an Expert offers free answers to your toughest academic and professional questions from over 65,000 verified experts. It’s trusted by millions of students each month with the majority of questions receiving an answer within 1 hour of being asked. If you ever need more than just an answer, Wyzant also offers personalized 1-on-1 sessions with experts that will work with you to help you understand whatever you’re trying to learn. Ask your own question for free: https://www.wyzant.com/resources/answ... Find a tutor for a 1-on-1 session: https://www.wyzant.com?utm_source=you... Subscribe to Wyzant on YouTube: https://www.youtube.com/subscription_...
