Calculus 2 Weiqi Zhou
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
determine monotonicities of following functions
show that there is precisly one real solution for
let , then
is monotonically non-decreasing，，，the result then follows.
show that for
let ，then
thus
show that has exactly one real root
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
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
find extrema and corresponding function values of following functions
find extrema and corresponding function values using second derivatives
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
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
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
the area above the curve is convex:
equivalently holds for any
if exists，then being convex
concave functions: revert inequlities above
the point where the convexity changes
if is twice differentiable，then vanishes at inflection Points
the opposite direction need not hold, e.g., or
determine convexities of following functions
show that for and