1, the state should be isomorphic, mainly for two variables: the equation is isomorphic up and down, and the two are combined into one Taishan shift.

f(x 1)-f(x2)/x 1-x2 & gt； k(x 1 & lt； x2)？ .

f(x 1)-f(x2)& lt； kx 1-kx2？ .

f(x 1)-kx 1 & lt； f(x2)-kxz .

Y=f(x)-kx is increasing function.

f(x 1)-f(x2)/x 1-x2 & lt； (k/x 1x 2(x 1 & lt； x2).

f(x 1)-f(x2)>k(x 1-x2)/x 1x 2 = k/x2-k/x 1 .

f(x 1)+k/x 1 & gt； F(x2)+k/x2→y=f(x)+k/x is a decreasing function.

It is a common deformation to sort out inequalities with two variables x 1, x2, or p, q with the same state. If the two sides of the inequality are structurally consistent (that is, isomorphic) after sorting, monotonicity is often implied (the sizes of the two variables need to be set in advance).

2, refers to cross-order isomorphism, with left and right logarithm. Isomorphism basic pattern.

Product type: aea≤blnb.

Same as right: elnea≤bInb→f(x)=xInx.

Same as left:: aea≤(lnb)elnb→f(x)=xex.

Right: a+Ina≤Inb+In(lnb)→f(x)=x+Inx.

3. Isomorphism needs to be scaled appropriately, and it is a flexible application of isomorphic thinking method to cut and paste isomorphic together. Scaling is also an ability. Using tangent scaling, local isomorphism is often needed. Using tangent scaling is just like using mean inequality, as long as the condition of taking equal sign holds. Master common scaling: (pay attention to the conditions of taking equal sign and common deformation).

ex≥x+ 1→ex- 1≥x→ex≥ex = ex≥E2/4x 2 .

ex≥ 1+x+x2/2 .

ex≤2+x/2-x(0≤x & lt； 2)。

ex≥ax+ 1(x≥0，0 & lt； a≤ 1).

It is very convenient to solve the mixed inequality problem of finger pairs, such as finding the range of parameters or proving inequalities. Of course, in practical use, it is often used in combination with tangent scaling or substitution method. It can be said that mastering these new favorites and common tangent inequalities will greatly reduce the difficulty of such problems.