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given a = (−1, 5), b = (x, 2), and c = (4, −6) and the sum of ab bc is to be a minimum, find the value of x.

User Rohit Ware
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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.

User Dmitrievanthony
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