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STD 12 - 5. continuity and differentiation — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 5. continuity and differentiation4 Marks
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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