Consider the family of cubic polynomials of the form f(x) = ax³ + bx² + cx + d where a, b, c, and d are constants. Note these constants are lowercase letters. (a) Determine the following: f'(x) = 3x² +2bx + c f''(x) = 6ax + 2b (b) Suppose f has the following characteristics on its graph: • an inflection point at x = 5 a critical point at x = 2 (i) Write an equation that conveys the conditions on a, b and c so that the graph of f has a critical point at x = 2. Your answer may contain a, b and c. Equation: (ii) Write an equation that conveys the conditions on a, b and c so that the graph of f has an inflection point at x = 5. Your answer may contain a, b and c. Equation: (iii) Suppose the leading coefficient of f is known to be 3. Determine the exact values of a, b, and c.

Consider the family of cubic polynomials of the form f(x) = ax³ + bx² + cx + d where a, b, c, and d are constants. Note these constants are lowercase letters. (a) Determine the following: f'(x) = 3x² +2bx + c f''(x) = 6ax + 2b (b) Suppose f has the following characteristics on its graph: • an inflection point at x = 5 a critical point at x = 2 (i) Write an equation that conveys the conditions on a, b and c so that the graph of f has a critical point at x = 2. Your answer may contain a, b and c. Equation: (ii) Write an equation that conveys the conditions on a, b and c so that the graph of f has an inflection point at x = 5. Your answer may contain a, b and c. Equation: (iii) Suppose the leading coefficient of f is known to be 3. Determine the exact values of a, b, and c.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.3: Zeros Of Polynomials

Problem 3E

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