4. Monotonicity, Convexity, Extrema

Calculus 2



Weiqi Zhou

Monotonicity and Derivatives

  • suppose that is continuous on ，differentiable in , if in there is further:

  • ，then monotonically increases
    ，then is monotonically non-decreasing

  • ，then monotonically decreases
    ，then is monotonically non-increasing

Exercises

  • determine monotonicities of following functions





Example

  • show that there is precisly one real solution for

  • let , then

  • is monotonically non-decreasing，，，the result then follows.

Example

  • show that for

  • let ，then

  • thus

Exercises

  • show that has exactly one real root

  • show that for

  • show that for

Monotonicity and Extrema

  • if is an extremum of , and is differentiable at , then

  • an extremum need not be differentiable: at

  • if is differentiable at an extremum，then monotonicities (and hence signs of ) differ at two sides

Monotonicity and Extrema

  • if the second derivative exists, then it can be used to decide on the type of the extremum

  • positive: a minimum；negative：a maximum；
    zero：undetermined

  • if is an extremum，and exists，then exists and equals ，consequently


  • the result follows after a discussion on the sign of over and respectively

Exercises

  • find extrema and corresponding function values of following functions





Exercises

  • find extrema and corresponding function values using second derivatives



Maximum/Minimum on an Interval

  • if is differentiable on , then its global maximum/minimum on is either a stationary point, or an undifferentiable point, or an end point of the interval

  • example: on

  • example: at

Exercises

  • find the maximum and minimum of on

  • minimize the sum of intercepts (on two axes) of tangent lines of

  • the cost of a beverage is for liters, the income is , find the that maximizes the profit

Convex Sets

  • a set is convex, if any line segment that connects two arbitrary points of is completely in

  • the line segment: for

  • being convex：, we have

Convex Functions

  • the area above the curve is convex:

  • equivalently holds for any

  • if exists，then being convex

  • concave functions: revert inequlities above

Inflection Point

  • the point where the convexity changes

  • if is twice differentiable，then vanishes at inflection Points

  • the opposite direction need not hold, e.g., or

Exercise

  • determine convexities of following functions





  • show that for and

Summary

  • Monotonicities

  • Extrema

  • Convexity