Answer is: 112
\(\int_{0}^{1} f(\lambda x) d \lambda=\operatorname{af}(x)\)
\(\lambda \mathrm{x}=\mathrm{t}\)
\(\mathrm{d} \lambda=\frac{1}{\mathrm{x}} \mathrm{dt}\)
\(\frac{1}{x} \int_{0}^{x} f(t) d t=a f(x)\)
\(\int_{0}^{x} f(t) d t=\operatorname{axf}(x)\)
\(f(x)=a\left(x f^{\prime}(x)+f(x)\right)\)
\((1-a) f(x)=a \cdot x f^{\prime}(x)\)
\(\frac{f^{\prime}(x)}{f(x)}=\frac{(1-a)}{a} \frac{1}{x}\)
\(\operatorname{\ell nf}(\mathrm{x})=\frac{1-\mathrm{a}}{\mathrm{a}} \operatorname{\ell n} \mathrm{x}+\mathrm{c}\)
\(\mathrm{x}=1, \mathrm{f}(1)=1 \Rightarrow \mathrm{c}=0\)
\(\mathrm{x}=16, \mathrm{f}(16)=\frac{1}{8}\)
\(\frac{1}{8}=(16)^{\frac{1-a}{a}} \Rightarrow-3=\frac{4-4 a}{a} \Rightarrow a=4\)
\(f(x)=x^{-\frac{3}{4}}\)
\(f^{\prime}(x)=-\frac{3}{4} x^{-\frac{7}{4}}\)
\(\therefore 16-\mathrm{f}^{\prime}\left(\frac{1}{16}\right)\)
\(=16-\left(-\frac{3}{4}\left(2^{-4}\right)^{-7 / 4}\right)\)
= 16 + 96 = 112
