Approximating Values of a Function Using Local Linearity and Linearization AB & BC

Near a point of differentiability a curve is nearly its tangent line, so $L(x) = f(a) + f'(a)(x - a)$ approximates $f(x)$ for $x$ near $a$. The center has to be both close to the target and exactly computable, which is why $\sqrt{27}$ is approximated from $a = 25$ as $5 + 0.1(2) = 5.2$. The $(x - a)$ is the piece that goes missing; without it the line has the right slope in the wrong place.

Whether the estimate is high or low is decided by concavity, not by whether the function is rising. A concave-down curve sits below its tangent, so the approximation overestimates, which is why $\sqrt{27} \approx 5.2$ exceeds $5.19615$; a concave-up curve sits above, so $e^{0.1} \approx 1.1$ falls short of $1.10517$. The claim that a tangent line always underestimates is half right and therefore wrong, and the sign of $f'$ settles nothing at all.