My favorite instructive math book I have read till now is the Cauchy-Schwarz Master Class written by Michael Steele et al.
Along the way the Cauchy-Schwarz inequality (obvious), ladder of power means, Hölder/Hilbert/Hardy inequalities, but also convexity arguments, Legendre transformation or Muirhead's inequality are introduced.
The authors manage to present the topic in a fun way and use algebraic and geometric arguments to show the beauty the field. The material is well-structured and leads through a well-structured chain of proofs. I never had so much fun in doing proofs, simple use of permutations and convexity leading to the Muirhead inequality and Schur's Majorization is quite an epiphany. The authors are clearly inspired by The Book.
(Given a class in calculus) The book is a perfect introduction for someone who wants to develop a love relationship to math. If I had an instructor as a pupil my view of math would have been quite different back then.
And it is of course useful for optimization introducing Jensen inequality, convex conjugates (and Hölder inequality), majorization and relationship between means in general.
To end an excerpt from the preface
To solve a problem is a very human undertaking, and more than a little mystery remains about how we best guide ourselves to the discovery of original solutions. Still, as George Pólya and others have taught us, there are principles of problem solving. With practice and good coaching we can all improve our skills. Just like singers, actors, or pianists, we have a path toward a deeper mastery of our craft.
Click on the image to go to Steele's website