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Question

# If x is real, then the maximum and minimum values of expression x2+14x+9x2+2x+3will be

A

4, - 5

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B

5, - 4

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C

- 4, 5

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D

- 4, - 5

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Solution

## The correct option is A 4, - 5 Let y=x2+14x+9x2+2x+3 ⇒ y(x2+2x+3)−x2−14x−9=0 ⇒ (y−1)x2+(2y−14)x+3y−9=0 For real x, its discriminant ≥ 0 i.e. 4(y−7)2−4(y−1)3(y−3)≥0 ⇒ y2+y−20≤0 or (y−4)(y+5)≤0 Now, the product of two factors is negative if these are of opposite signs. So following two cases arise: Case 1: y−4≥0 or y≥4 y+5≤0 or y≤−5 This is not possible. Case 2: y−4≤0 or y≤4 y+5≥0 or y≥−5 Both of these are satisfied if -5≤y≤4 hence maximum value of y is 4 and minimum value is -5

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