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The ninth grade first volume mathematics ability training and test answer.
Senior three students thought about this problem for a long time, and finally ~~(x = 1 1/4, the minimum value is 8.2462).

Method 1: First, deform your formula. The two radicals are (X-2) 2+9 and (X-4) 2+25 respectively.

Then consider these two circles whose centers are (2, 3) and (4, 5) respectively, and the points on the circles move on the X axis (because y=0), the problem becomes to find the minimum sum of the radii of the two circles;

Furthermore, this is the shortest path problem, which can be transformed into finding the sum of the distances from a point on the X axis to (2, -3) and (4, 5). According to the shortest straight line between two points, the sum of the minimum radii is the distance between two points (2, -3) and (4, 5), and the root number is 68 =8.2462. Then, a linear equation is established according to these two points.

Note: It is suggested to draw a rectangular coordinate on the paper, and it will be understood that (2,3) and (2,3) are symmetrical about X axis.

Method 2: I made it with matlab. By differentiation, here is the program.

& gt& gtsyms x;

y=((x-2)^2+9)^(0.5)+((x-4)^2+25)^(0.5);

k = diff(y);

[x]= solve (k)

X= 1 1/4 and the minimum value of y is 8.2462.

(matlab special mathematical software, learned when I entered the university)