Let f x = x 2 + 2 p x + 2 q 2 and g x = - x 2 - 2 q x + p 2 (where q ≠ 0 ). If x ∈ R and the the minimum value of f x is equal to the maximum value of g x , then the value of p 2 q 2 is equal to

Question

Let f x = x 2 + 2 p x + 2 q 2 and g x = - x 2 - 2 q x + p 2 (where q ≠ 0 ). If x ∈ R and the the minimum value of f x is equal to the maximum value of g x , then the value of p 2 q 2 is equal to,

MARKS App · Accepted Answer

Since the minimum/maximum value of a quadratic expression a x 2 + b x + c is given by 4 b c - b 2 4 a So, f m i n = 2 q 2 - p 2 , g m i n = p 2 + q 2 Now, 2 q 2 - p 2 = p 2 + q 2 ⇒ q 2 = 2 p 2 Hence, p 2 q 2 = 1 2 = 0 . 5

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