# CAT 2020 Question Paper | Quants Slot 2

###### CAT Previous Year Paper | CAT Quants Questions | Question 14

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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

1. 16
2. 1
3. 4
4. 0

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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

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"

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