Grand Review - Advanced Differentiation, Function Analysis and De L'Hopital's Theorem.

Differentiation by considering every other variable as a function of x, differentiation rules apply.

Using the logarithm change of base rule, we convert to base-e and differentiate:

Point where , around which it changes sign

Point where derivative tends to infinity from both sides.

Point where derivative tends to infinities of opposite signs.

Point where , around which it changes sign. The function switches concavity at an inflection point

A function is convex if the and concave if

You may be able to find the limit for an indeterminate form by instead taking the same limit but of the ratio of the derivatives of the functions instead.

It's time for a crossover episode yet again.
Calculus Interlude - The Weierstrass or Extreme Value Theorem