Question
The number of real-valued solutions of the equation \({2^x} + {2^{ - x}} = 2 - {(x - 2)^2}\) is
- infinite
- 1
- 0
- 2
Option: 3
Solution:
Solution:
\({2^x} + {2^{ - x}} = 2 - {(x - 2)^2}\)
The minimum value of \({2^x} + {2^{ - x}}\) is 2 when \(x = 0\)
But \(x = 0;2 - {(x - 2)^2} = - 2\)
The maximum value of \(2 - {(x - 2)^2}\) is 2 when \(x = 2\)
But \(x = 2\quad {2^x} + {2^{ - x}} = \frac{{17}}{4}\)
Hence there is no value of \(x,{2^x} + {2^{ - x}} = 2 - {(x - 2)^2}\)
The number of solutions is 0
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