Let f : [0,1] [0,1] be a continuous function, then the equation f… - Vidyadip

MCQ

Let $f : [0,1] \to [0,1]$ be a continuous function, then the equation $f(x) = x$

A
may not have any solution in $[0,1]$
B
must have exactly one solution in $[0,1]$
✓
must have atleast one solution in $[0,1]$
D
must have atleast two solutions in $[0,1]$

✓

Answer

Correct option: C.

must have atleast one solution in $[0,1]$

c
Define new function $\mathrm{g}(\mathrm{x})=f(\mathrm{x})-\mathrm{x}$ in $[0,1]$

then $g(1)=f(1)-1 \leq 0 $ and $ g(0)=f(0) \geq 0$

$\therefore $ equation $\mathrm{g}(\mathrm{x})=0$ has atleast one root in $[0,1]$

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