Let be a twice differentiable non-negative function such that Then the mean of is equal to _________ . [2026]

Question

Let $f$ be a twice differentiable non-negative function such that $(f {(x))}^{2} = 25 + \int_{0}^{x} ((f {(t))}^{2} + (f' {(t))}^{2}) d t .$ Then the mean of $f (\log_{e} (1)), f (\log_{e} (2)), \dots, f (\log_{e} (625))$ is equal to _________ . [2026]

Smart Achievers · Accepted Answer

(1565)

$2 f (x) f' (x) = f^{2} (x) + (f' {(x))}^{2}$

$\Rightarrow (f (x) - f' {(x))}^{2} = 0$

$\Rightarrow f (x) = f' (x)$

$\Rightarrow l n (f (x)) = x + c \Rightarrow f (x) = c' e^{x}$

$f (0) = 5 \Rightarrow f (x) = 5 e^{x}$

$Mean = \frac{f (l n 1) + f (l n 2) + \dots + f (l n 625)}{625}$

$= \frac{5 (1 + 2 + \dots + 625)}{625} = 1565$

Solution

Q.
Let $f$ be a twice differentiable non-negative function such that $(f {(x))}^{2} = 25 + \int_{0}^{x} ((f {(t))}^{2} + (f' {(t))}^{2}) d t .$ Then the mean of $f (\log_{e} (1)), f (\log_{e} (2)), \dots, f (\log_{e} (625))$ is equal to _________ . [2026]

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Solution

Q. Let f be a twice differentiable non-negative function such that (f(x))2=25+∫0x((f(t))2+(f'(t))2) dt. Then the mean of f(loge(1)), f(loge(2)), …, f(loge(625)) is equal to _________ . [2026]

Ans. (1565) 2f(x)f'(x)=f2(x)+(f'(x))2 ⇒(f(x)-f'(x))2=0 ⇒f(x)=f'(x) ⇒ln(f(x))=x+c⇒f(x)=c'ex f(0)=5⇒f(x)=5ex Mean=f(ln1)+f(ln2)+…+f(ln625)625 =5(1+2+…+625)625=1565

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Q.
Let $f$ be a twice differentiable non-negative function such that $(f {(x))}^{2} = 25 + \int_{0}^{x} ((f {(t))}^{2} + (f' {(t))}^{2}) d t .$ Then the mean of $f (\log_{e} (1)), f (\log_{e} (2)), \dots, f (\log_{e} (625))$ is equal to _________ . [2026]