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有些数学题具有自己的特点,在解题中如果能够充分利用这些特点,会使解法简捷。下面举数例说明。 例1.如果x、y满足3x+2y=6,求x~2+y~2+6x+8y+25的最小值。 [分析]这类题通常的解法是利用代入法将二元函数化为一元二次函数,再利用一元二次函数的性质求出极值。但本题有自己的特点。分析多项式x~2+y~2+6x+8y+25,可以看到它可配方成(x+3)~2+(y+4)~2。如果设f(x,y)=(x+3)~2+(y+4)~2,那么,f(x,y)就是坐标平面内一点到点N(-3,-4)距离的平方。由于x、y满足3x+2y=6,所以
Some mathematics problems have their own characteristics. If you can make full use of these features in the problem solving, it will make the solution simple. The following shows a few examples. Example 1. If x and y satisfy 3x+2y=6, find the minimum value of x~2+y~2+6x+8y+25. [Analysis] The usual solution to this kind of problem is to use the substitution method to convert the binary function into a quadratic function, and then use the properties of the unary quadratic function to find the extremum. But this question has its own characteristics. Analyzing the polynomial x~2+y~2+6x+8y+25, we can see that it can be formulated as (x+3)~2+(y+4)~2. If we set f(x,y)=(x+3)~2+(y+4)~2, then f(x,y) is the distance from point one to point N(-3,-4) in the coordinate plane. square. Since x and y satisfy 3x+2y=6,