Why does the second derivative test work

Lines 2. Distance Between Two Points; Circles 3. Functions 4. The slope of a function 2. An example 3. Limits 4. The Derivative Function 5. Properties of Functions 3 Rules for Finding Derivatives 1.

The Power Rule 2. Linearity of the Derivative 3. The Product Rule 4. The Quotient Rule 5. The Chain Rule 4 Transcendental Functions 1. Trigonometric Functions 2. A hard limit 4. Derivatives of the Trigonometric Functions 6. Exponential and Logarithmic functions 7. Derivatives of the exponential and logarithmic functions 8. Implicit Differentiation 9. Inverse Trigonometric Functions Limits revisited Hyperbolic Functions 5 Curve Sketching 1.

Maxima and Minima 2. The first derivative test 3. The second derivative test 4. Concavity and inflection points 5. Optimization 2.

Related Rates 3. Newton's Method 4. Linear Approximations 5. The Mean Value Theorem 7 Integration 1. Two examples 2. The Fundamental Theorem of Calculus 3. Some Properties of Integrals 8 Techniques of Integration 1. Substitution 2. Powers of sine and cosine 3.

Trigonometric Substitutions 4. Integration by Parts 5. Rational Functions 6. Numerical Integration 7. Additional exercises 9 Applications of Integration 1. Area between curves 2. Distance, Velocity, Acceleration 3. Volume 4. Average value of a function 5. Work 6. Center of Mass 7. Kinetic energy; improper integrals 8. Probability 9. Arc Length Polar Coordinates 2. Slopes in polar coordinates 3. Areas in polar coordinates 4. Parametric Equations 5. Calculus with Parametric Equations 11 Sequences and Series 1.

Sequences 2. Series 3. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. I haven't seen this asked much, if at all, and a lot of college books in my knowledge never provide the reasoning behind why the second derivative test even works? Otherwise the test is inconclusive. In the first three cases the "second derivative test" gives a factual result. The reason why regular college textbooks don't justify why this works, is because the proof is too advanced.

You cannot get 20 year old college kids, who are taking their 3rd course in Calculus, to truly understand it. You need to dig deeper and check out Real Analysis textbooks.

Find a Real Analysis textbook, go to the multivariate part, and you'll find a proof. A hint on maybe finding some intuitive answer about why this works: on YouTube there's a playlist called Khan Academy multivariate calculus which is taught by Grant Sanderson, the man behind the 3Blue1Brown YouTube channel. Search for the video about the second derivative test, and you will probably have your answer. Grant likes to give intuitive explanations about why things are the way they are instead of just hammering the proof on your soul.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why does the second derivative test in multivariable functions work? Ask Question. Asked 1 year, 1 month ago. Active 1 year, 1 month ago. Viewed times.