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29 Jul 2024, 04:22
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What is the minimum value of \(f(x) = |x + 4| + |x + 1| + |x - 2|\)?
A. 0
B. 1
C. 4
D. 6
E. 9
A. 0
B. 1
C. 4
D. 6
E. 9
29 Jul 2024, 04:22
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Official Solution:
What is the minimum value of \(f(x) = |x + 4| + |x + 1| + |x - 2|\)?
A. 0
B. 1
C. 4
D. 6
E. 9
APPROACH 1:
To find the minimum value of \(f(x) = |x + 4| + |x + 1| + |x - 2|\), we need to analyze the critical points where the absolute values change, which are at \(x = -4\), \(x = -1\), and \(x = 2\). We will evaluate the function around these points.
1. For \(x ≤ -4\):
\(f(x) = -(x + 4) - (x + 1) - (x - 2) =\)
\(= -3x - 3\)
In this range, the minimum value of \(f(x)\) occurs when \(x = -4\): \(f(-4) = -3(-4) - 3 = 12 - 3 = 9\)
2. For \(-4 < x < -1\):
\(f(x) = (x + 4) - (x + 1) - (x - 2) =\)
\(= -x + 5\)
In this range, the minimum value of \(f(x)\) occurs as \(x\) approaches -1 from the left: \(f(x)\) approaches \(-(-1) + 5 = 1 + 5 = 6\)
3. For \(-1 ≤ x < 2\):
\(f(x) = (x + 4) + (x + 1) - (x - 2) =\)
\(= x + 7\)
In this range, the minimum value of \(f(x)\) occurs when \(x = -1\): \(f(-1) = -1 + 7 = 6\)
4. For \(x ≥ 2\):
\(f(x) = (x + 4) + (x + 1) + (x - 2) =\)
\(= 3x + 3\)
In this range, the minimum value of \(f(x)\) occurs when \(x = 2\): \(f(2) = 3(2) + 3 = 6 + 3 = 9\).
Comparing these values, the minimum value of \(f(x)\) is 6, which occurs at \(x = -1\).
APPROACH 2:
\(|x + 4| + |x + 1| + |x - 2|\) represents the sum of the distances of \(x\) from -4, -1, and 2.
--------(-4)------------(-1)------------(2)--------
We can notice that the further we go from -4 to the left, or from 2 to the right, the total distance increases. Hence, the minimum distance should occur when \(x\) is somewhere between -4 and 2. In that range, when we move from -1 in either direction, the distance from two of the points increases while the distance from the third one decreases, resulting in a net increase in the total distance. For example, at \(x = 0\), the distances are 4, 1, and 2, totaling 7 units. At \(x = 1\), the distances are 5, 2, and 1, totaling 8 units. Thus, exactly at \(x = -1\), we'd get the smallest sum of the distances: 3, 0, and 3, which total 6 units.
--------(-4)------------(-1)------------(2)--------
Therefore, the minimum value of the function \(f(x)\) occurs at \(x = -1\), and the minimum value is 6.
Answer: D
What is the minimum value of \(f(x) = |x + 4| + |x + 1| + |x - 2|\)?
A. 0
B. 1
C. 4
D. 6
E. 9
APPROACH 1:
To find the minimum value of \(f(x) = |x + 4| + |x + 1| + |x - 2|\), we need to analyze the critical points where the absolute values change, which are at \(x = -4\), \(x = -1\), and \(x = 2\). We will evaluate the function around these points.
1. For \(x ≤ -4\):
\(f(x) = -(x + 4) - (x + 1) - (x - 2) =\)
\(= -3x - 3\)
In this range, the minimum value of \(f(x)\) occurs when \(x = -4\): \(f(-4) = -3(-4) - 3 = 12 - 3 = 9\)
2. For \(-4 < x < -1\):
\(f(x) = (x + 4) - (x + 1) - (x - 2) =\)
\(= -x + 5\)
In this range, the minimum value of \(f(x)\) occurs as \(x\) approaches -1 from the left: \(f(x)\) approaches \(-(-1) + 5 = 1 + 5 = 6\)
3. For \(-1 ≤ x < 2\):
\(f(x) = (x + 4) + (x + 1) - (x - 2) =\)
\(= x + 7\)
In this range, the minimum value of \(f(x)\) occurs when \(x = -1\): \(f(-1) = -1 + 7 = 6\)
4. For \(x ≥ 2\):
\(f(x) = (x + 4) + (x + 1) + (x - 2) =\)
\(= 3x + 3\)
In this range, the minimum value of \(f(x)\) occurs when \(x = 2\): \(f(2) = 3(2) + 3 = 6 + 3 = 9\).
Comparing these values, the minimum value of \(f(x)\) is 6, which occurs at \(x = -1\).
APPROACH 2:
\(|x + 4| + |x + 1| + |x - 2|\) represents the sum of the distances of \(x\) from -4, -1, and 2.
--------(-4)------------(-1)------------(2)--------
We can notice that the further we go from -4 to the left, or from 2 to the right, the total distance increases. Hence, the minimum distance should occur when \(x\) is somewhere between -4 and 2. In that range, when we move from -1 in either direction, the distance from two of the points increases while the distance from the third one decreases, resulting in a net increase in the total distance. For example, at \(x = 0\), the distances are 4, 1, and 2, totaling 7 units. At \(x = 1\), the distances are 5, 2, and 1, totaling 8 units. Thus, exactly at \(x = -1\), we'd get the smallest sum of the distances: 3, 0, and 3, which total 6 units.
--------(-4)------------(-1)------------(2)--------
Therefore, the minimum value of the function \(f(x)\) occurs at \(x = -1\), and the minimum value is 6.
Answer: D
29 Dec 2025, 00:39
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Hi Bunuel, In this solution, how did you determine when x would be ≤ or just <. I am referring to −4<x<−1 and −1≤x<2 in the second and third respective stages.
