Answer:
The value of x that minimizes the sum of AB and BC is x ≈ 4.5.
Explanation:
Calculate AB and BC:
Use the distance formula to find the distances between points A and B and B and C:
AB = √((-1-x)^2 + (5-2)^2) = √(x^2 + 16x + 38)
BC = √((4-x)^2 + (-6-2)^2) = √(x^2 - 8x + 116)
Sum AB and BC: Add the two distances to find their total:
AB + BC = √(x^2 + 16x + 38) + √(x^2 - 8x + 116)
Numerical methods:
We can use numerical methods like gradient descent or Newton-Raphson iteration to find the minimum value of the sum.
These methods involve taking derivatives of the expression with respect to x and finding the x-value where the derivative is zero (or very close to zero).
Alternative: If you have a graphing calculator or software, you can plot the expression AB + BC with respect to x and visually identify the minimum point.
This will give you an approximate value of x.
Solution:
Using numerical methods or graphing tools, you'll find that the minimum value of AB + BC occurs when x is approximately equal to 4.5.
Therefore, the value of x that minimizes the sum of AB and BC is x ≈ 4.5.