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Question 14 : Let f(x) = x² + ax + b and g(x) = f(x + 1) - f(x - 1). If f(x) ≥ 0 for all real x, and g(20) = 72, then the smallest possible value of b is

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Video Explanation

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Explanatory Answer

f(x+1) = f(x+1)² + a(x+1) + b
f(x-1) = f(x-1)² + a(x-1) + b
g(x) = (x+1)² + a(x+1) + b - (x-1)² - a(x-1) – b
= 4x + 2a
g(20) = 72 = 4(20) + 2a
a = - 4
x2 - 4x + b ≥ 0
(x-2)² +b – 4 ≥ 0
Value of b is at least 4

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The question is "Let f(x) = x² + ax + b and g(x) = f(x + 1) - f(x - 1). If f(x) ≥ 0 for all real x, and g(20) = 72, then the smallest possible value of b is"

Hence, the answer is, "4"