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)