The number of real-valued solutions of the equation 2x + 2-x = 2 – (x – 2)² is
Explanation:
2x + 2-x is of the form y + 1/y Minimum value the expression can take is 2. Hence, 2x + 2-x ≥ 2
Now, 2 - (x - 2)2 We know, (x - 2)2 ≥ 0 ∴ 2 - (x - 2)2 ≤ 2 (From 2 we are subtracting a non-negative number) Maximum value this expression can have is 2.
The only possibility is both sides are = 2 LHS = 2x + 2-x = 0 This is possible only when x = 0.
When x = 0, RHS = 2 - (x - 2)2 = 2 - 22 = - 2 Hence, at x = 0, LHS ≠ RHS. ∴ There is no solution possible. Hence, option (a).
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