The number of continuous functions f:[0,1]→R that satisfy 1∫0xf(x)dx=13+141∫0(f(x))2dx
A
0
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B
1
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C
2
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D
Infinity
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Solution
The correct option is B1 1∫0xf(x)dx=13+141∫0(f(x))2dx⇒−13=1∫0⎛⎝(f(x)2)2−xf(x)⎞⎠dx⇒1∫0(f(x)2−x)2dx−1∫0x2dx=−13⇒1∫0(f(x)2−x)2dx=0 ⇒(f(x)2−x)2=0 ∴f(x)=2x Only one function exists.