(a) Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$.
(b) Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.
ridgers wrote:
(a) Find the minimal distance between the points of the graph of the function $y=ln x$ from the line $y=x$
$f(x)=\ln x$ and $g(x)=x$ are $C_{\infty}$ function from $\mathbb R^+\to\mathbb R$ with no intersection and so the minimal distance is obtained at a point where $f'(x)=1$ and so is $x=1$ and so this distance is $\boxed{\frac{\sqrt 2}2}$
ridgers wrote:
[(b) Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.
$f(x)=\ln x$ and $g(x)=e^x$ have symetric graph (with symetry axe $y=x$) and so, using the previous result, minimal distance is twice the previous, and so is $\boxed{\sqrt 2}$
(a) The tangent line to the curve $y=\log x$ at the point $x=t$ has equation $y=\frac x t+\log t-1$; its slope is the same of $y=x$ for $t=1$ (which is the point we're actually searching for since $\log x$ is concave on $(0, \infty]$), where the distance between the two curves is $\frac 1 \sqrt 2$.
(b) The graphs of the two functions are symmetric about the bisector of the first quadrant, so the minimal distance between them, by the previous point, is $\sqrt 2$.