Answer to Question #254384 in Calculus for Hasni

Question

Let f be a differentiable function that is positive everywhere and such that

f′(x)=−f(x)ln(f(x))for allx∈ℝ.

f′(x)=−f(x)ln⁡(f(x))for allx∈R.

(a) Use logarithmic differentiation for the function g(x)=(f(x))^e^x to show that g′(x)=0 for all x∈ℝ

(b) Then, using the fact that any function whose derivative vanishes over some interval must be constant over that interval, deduce that f(x)=e^(e^-x)* ln C for some positive constant C

Answer 1

"\\mathrm{(}a\\mathrm{)}\\ \\ g\\left(x\\right)={\\left(f\\left(x\\right)\\right)}^{e^x} \\\\ \n \\space\nTaking\\ \\ \\mathrm{"ln"}\\ \\ of\\ \\ both\\ \\ side, \\\\ \n \\\\\n\\\\ \n\\mathrm{ln}\\left(g\\left(x\\right)\\right)=e^x\\ \\mathrm{ln}\\left(f\\left(x\\right)\\right) \\\\ \n \\\\ \nDifferentiate\\ Both\\ \\ Side\\ \\ of\\ \\ the\\ \\ equation, \\\\ \n \\\\ \n\\frac{g'\\left(x\\right)}{g\\left(x\\right)}\\ \\ =\\ e^x\\ \\mathrm{ln}\\left(f\\left(x\\right)\\right)+e^x\\ \\frac{f'\\left(x\\right)}{f\\left(x\\right)} \\\\ \n \\\\ \ng'\\left(x\\right)=g\\left(x\\right)\\left(e^x\\ \\mathrm{ln}\\left(f\\left(x\\right)\\right)+e^x\\ \\frac{f'\\left(x\\right)}{f\\left(x\\right)}\\right) \\\\ \n \\\\ \nBut\\ \\ given\\ \\ that\\ \\ \\ f'\\left(x\\right)=-f\\left(x\\right){}{}{}\\ \\ \\mathrm{ln}\\left(f\\left(x\\right)\\right) \\\\ \n \\\\ \nTherefore,\\ \\ g'\\left(x\\right)=g\\left(x\\right)\\left(e^x\\ \\mathrm{ln}\\left(f\\left(x\\right)\\right)-e^x\\ \\frac{f\\left(x\\right){}{}{}\\ \\ \\mathrm{ln}\\left(f\\left(x\\right)\\right)}{f\\left(x\\right)}\\right) \\\\ \n \\\\ \ng'\\left(x\\right)=g\\left(x\\right)\\left(e^x\\ \\mathrm{ln}\\left(f\\left(x\\right)\\right)-e^x\\ \\mathrm{ln}\\left(f\\left(x\\right)\\right)\\right)\\ \\ =\\ \\ 0 \\\\ \n \\\\ \n\\left(b\\right)f'\\left(x\\right)=-f\\left(x\\right){}{}{}\\ \\ \\mathrm{ln}\\left(f\\left(x\\right)\\right) \\\\ \n \\\\ \n\\frac{df\\left(x\\right)}{ \\begin{array}{l}\nf\\left(x\\right){}{}{}\\ \\ \\mathrm{ln}\\left(f\\left(x\\right)\\right) \\\\ \n \\end{array}\n}=-dx \\\\ \n \\\\ \n\\int{\\frac{d\\left(f\\left(x\\right)\\right)}{f\\left(x\\right){}{}{}\\ \\ \\mathrm{ln}\\left(f\\left(x\\right)\\right)}=-x+\\mathrm{ln}C} \\\\ \n \\\\ \n\\mathrm{ln}\\left(\\mathrm{ln}\\left(f\\left(x\\right)\\right)\\right)=-x+\\mathrm{ln}C \\\\ \n \\\\ \n\\mathrm{ln}\\left(f\\left(x\\right)\\right)=e^{-x+C} \\\\ \n \\\\ \nf\\left(x\\right)=e^{\\mathrm{(}e^{-x}\\mathrm{)ln}C}"