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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Let $f$ be a twice differentiable non-negative function such that
$(f(x))^2 = 25 + \int_0^x \big((f(t))^2 + (f'(t))^2\big),dt$.
Then the mean of $f(\log 1),; f(\log 2),; \ldots,; f(\log 625)$ is equal to ______.
Solution
$2f(x)f'(x) = f(x) + (f'(x))^2$
$\Rightarrow (f(x) - f'(x))^2 = 0$
$\Rightarrow f(x) = f'(x)$
$\Rightarrow \ln f(x) = x + c \Rightarrow f(x) = ce^x$
$f(0) = 5 \Rightarrow f(x) = 5e^x$
Mean
$= \frac{f(\ln 1) + f(\ln 2) + \cdots + f(\ln 625)}{625}$
$= \frac{5(1 + 2 + \cdots + 625)}{625}$
$= \frac{5 \cdot \frac{625 \cdot 626}{2}}{625} = 1565$
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