Question
Let \(f(x) = {x^2} + ax + b\) and \(g(x) = f(x + 1) - f(x - 1)\). If \(f(x) \ge 0\) for all real x, and \(g(20) = 72\), then the smallest possible value of b is
- 1
- 16
- 0
- 4
Option: 4
Solution:
Solution:
\(f(x) = {x^2} + ax + b\)
\(g(x) = f(x + 1) - f(x - 1)\)
\( = \left\{ {{{(x + 1)}^2} + a(x + 1) + b} \right\} - \left\{ {{{(x - 1)}^2} + a(x - 1) + b} \right\}\)
\(g(x) = 4x + 2a\)
\(g(20) = 72\)
\(80 + 2a = 72 \Rightarrow a = - 4\)
\(\therefore f(x) = {x^2} - 4x + b\)
\(f(x) = {(x - 2)^2} + b - 4\)
when \(b \ge 4f(x) \ge 0\) for all x
\(\therefore \) The minimum value of b is 4
CAT 2020 Quant questions with Solutions
- CAT 2020 Quant Question 1
- CAT 2020 Quant Question 2
- CAT 2020 Quant Question 3
- CAT 2020 Quant Question 4
- CAT 2020 Quant Question 5
- CAT 2020 Quant Question 6
- CAT 2020 Quant Question 7
- CAT 2020 Quant Question 8
- CAT 2020 Quant Question 9
- CAT 2020 Quant Question 10
- CAT 2020 Quant Question 11
- CAT 2020 Quant Question 12
- CAT 2020 Quant Question 13
- CAT 2020 Quant Question 14
- CAT 2020 Quant Question 15
- CAT 2020 Quant Question 16
- CAT 2020 Quant Question 17
- CAT 2020 Quant Question 18
- CAT 2020 Quant Question 19
- CAT 2020 Quant Question 20
- CAT 2020 Quant Question 21
- CAT 2020 Quant Question 22
- CAT 2020 Quant Question 23
- CAT 2020 Quant Question 24
- CAT 2020 Quant Question 25
- CAT 2020 Quant Question 26
- CAT 2020 Quant Question 27
- CAT 2020 Quant Question 28
- CAT 2020 Quant Question 29
- CAT 2020 Quant Question 30
- CAT 2020 Quant Question 31
- CAT 2020 Quant Question 32
- CAT 2020 Quant Question 33
- CAT 2020 Quant Question 34
- CAT 2020 Quant Question 35
- CAT 2020 Quant Question 36
- CAT 2020 Quant Question 37
- CAT 2020 Quant Question 38
- CAT 2020 Quant Question 39
- CAT 2020 Quant Question 40
- CAT 2020 Quant Question 41
- CAT 2020 Quant Question 42
- CAT 2020 Quant Question 43
- CAT 2020 Quant Question 44
- CAT 2020 Quant Question 45
- CAT 2020 Quant Question 46
- CAT 2020 Quant Question 47
- CAT 2020 Quant Question 48
- CAT 2020 Quant Question 49
- CAT 2020 Quant Question 50
- CAT 2020 Quant Question 51
- CAT 2020 Quant Question 52
- CAT 2020 Quant Question 53
- CAT 2020 Quant Question 54
- CAT 2020 Quant Question 55
- CAT 2020 Quant Question 56
- CAT 2020 Quant Question 57
- CAT 2020 Quant Question 58
- CAT 2020 Quant Question 59
- CAT 2020 Quant Question 60
- CAT 2020 Quant Question 61
- CAT 2020 Quant Question 62
- CAT 2020 Quant Question 63
- CAT 2020 Quant Question 64
- CAT 2020 Quant Question 65
- CAT 2020 Quant Question 66
- CAT 2020 Quant Question 67
- CAT 2020 Quant Question 68
- CAT 2020 Quant Question 69
- CAT 2020 Quant Question 70
- CAT 2020 Quant Question 71
- CAT 2020 Quant Question 72
- CAT 2020 Quant Question 73
- CAT 2020 Quant Question 74
- CAT 2020 Quant Question 75
- CAT 2020 Quant Question 76
- CAT 2020 Quant Question 77
- CAT 2020 Quant Question 78