I am new to Taylor series as a whole and was wondering if someone with a bit more background could validate my thought process in answering the following question.
Question: Does the tangent line to $y=x \ln x$, at any point $p>0$ over-estimate or under-estimate $x \ln x$
Response: $T_1(x) = p\ln(p) + [\ln(p) + 1][x-p]$ As such, we consider the next term in the Taylor series: $${f^{2}(p)}/2! = 1/2p$$
As per Taylor's theorem, $f(x) = T_1(x) + [f^2(p)/2!][x-p]^2$
The term $[f^2(p)/2!][x-p]^2$ is ALWAYS POSITIVE for all p>0 $->$ Implying that the remainder term is positive. Hence the tangent line approximation for $x\ln(x)$, which is equal to $T_1(x)$ is an under-estimation for $y=x\ln(x)$