y=e to the power of x has a maximum point in [0, ∞], but no extreme point

y=3x-x to the third power

All four types of discontinuity points can be used to reach extreme points, as long as the left and right sides are defined.

It does not have to be differentiable at the extreme point, as long as it is differentiable in the neighborhood of this point

Necessary conditions (Fermat)

first sufficient condition

second sufficient condition

third sufficient condition

You can also use the definition to identify

convex

Concave

The inflection point only needs to be continuous

Concave and convex in no particular order

The inflection point is on the curve

Second derivative > 0, concave

Second derivative <0, convex

necessary conditions

first sufficient condition

second sufficient condition

third sufficient condition

You can also use the definition to identify

5.5 Proof of monotonicity

The second derivative > 0 near a point

Identify extreme values ​​according to the definition of extreme values

no definition point

Define the endpoints of the interval

piecewise function piecewise point

limx tends to x0 =∞ (or limx tends to x0-=∞), then x=x0 is a vertical asymptote

limx tends to ∞=y1, then y=y1 is a horizontal asymptote

limx tends to -∞=y2, then y=y2 is a horizontal asymptote

If = the same asymptote, y = y0 is a horizontal asymptote

limx tends to ∞, limf(x)/x=a1 (a cannot=0.) lim[f(x)-a1x]=b1

For y=a1x b1 is an oblique asymptote

similar

step

The same method of finding limits★★

x tends to infinity, 1/x=0

lne to the power of x × (e to the power of -x 1) = x ln (e to the power of -x 1)

1-(1-1/n) to the kth power~k/n

The point where the first derivative is zero

Points where the first derivative does not exist, underivable points

endpoint

Big loophole

stationary point

non-derivable point

The right limit of the left endpoint, the left limit of the right endpoint

If you encounter practical problems of finding the maximum and minimum values, first establish the objective function, and then convert it into an optimal value problem after determining the definition interval.

In particular, if the actual problem under consideration has a maximum or minimum value. The objective function has a unique extreme point, so it must be the maximum point. Find the n largest term under nth root.

The undefined point of f(x)

Point f'(x)=0

Points where f'(x) does not exist

Point f′′(x)=0

Worth doing several times

Implicit function extreme value

5.2

The arithmetic mean is greater than the geometric mean

common factor method

Examination of computing ability

First derivative = 0

The first derivative does not exist

Second derivative = 0

The second derivative does not exist

Pay attention to the approximation