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5. Continuity and Differentiation
medium
વિધેય $f(x) = x(x + 3){e^{ - (1/2)x}}$ એ અંતરાલ $[-3, 0]$ માં રોલના પ્રમેયનું પાલન કરે છે તો $c$ ની કિમંત મેળવો.
A
$0$
B
$-1$
C
$-2$
D
$-3$
Solution
(c) To determine $'c' $ in Rolle's theorem, $f'(c) = 0$.
Here $f'(x) = ({x^2} + 3x){e^{ – (1/2)x}}.\left( { – \frac{1}{2}} \right) + (2x + 3){e^{ – (1/2)x}}$
$ = {e^{ – (1/2)x}}\left\{ { – \frac{1}{2}({x^2} + 3x) + 2x + 3} \right\}$
$ = – \frac{1}{2}{e^{ – (x/2)}}\{ {x^2} – x – 6\} $
$\therefore f'(c) = 0 \Rightarrow {c^2} – c – 6 = 0 \Rightarrow c = 3,\, – 2,$
But $c = 3 \notin [ – 3,\,0].$
Standard 12
Mathematics
