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The minimum value of the expression |x+3|+|x-3|+|x-6|+|x-5|+|x+5|
- a)21
- b)19
- c)24
- d)25
Correct answer is option 'B'. Can you explain this answer?
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The minimum value of the expression |x+3|+|x-3|+|x-6|+|x-5|+|x+5|a)21b...
Let f(x) = |x+3|+|x-3|+|x-6|+|x-5|+|x+5|
f(x) is a linear equation in x at all times. It will be a union of straight lines with inflection points at -3, 3, 6, 5 and -5. Hence, the minimum value of the expression occurs at one of these inflection points. We will calculate the value of f(x) at each of these points and then find out the least possible value of f(x)
f(-3) = 25
f(3) = 19
f(6) = 24
f(5) = 21
f(-5) = 31
19 is the least value.
f(x) is a linear equation in x at all times. It will be a union of straight lines with inflection points at -3, 3, 6, 5 and -5. Hence, the minimum value of the expression occurs at one of these inflection points. We will calculate the value of f(x) at each of these points and then find out the least possible value of f(x)
f(-3) = 25
f(3) = 19
f(6) = 24
f(5) = 21
f(-5) = 31
19 is the least value.
Most Upvoted Answer
The minimum value of the expression |x+3|+|x-3|+|x-6|+|x-5|+|x+5|a)21b...
To find the minimum value of the given expression, we need to find the values of x that minimize each factor within the absolute value signs.
Let's analyze each factor separately:
1. |x-3|:
- When x < 3,="" the="" factor="" becomes="" -(x-3)="" />
- When x ≥ 3, the factor becomes x-3.
- So, the minimum value of this factor is 0 when x = 3.
2. |x-5|:
- When x < 5,="" the="" factor="" becomes="" -(x-5)="" />
- When x ≥ 5, the factor becomes x-5.
- So, the minimum value of this factor is 0 when x = 5.
3. |x-6|:
- When x < 6,="" the="" factor="" becomes="" -(x-6)="" />
- When x ≥ 6, the factor becomes x-6.
- So, the minimum value of this factor is 0 when x = 6.
4. |x-5|:
- As we have already determined, the minimum value of this factor is 0 when x = 5.
5. |x-3|:
- As we have already determined, the minimum value of this factor is 0 when x = 3.
Now, let's consider the remaining factor |x^3|:
- Since this factor involves x raised to an odd power, it is always non-negative. Therefore, its minimum value is also 0.
As all the factors have a minimum value of 0, the minimum value of the given expression is 0.
Hence, the correct answer is option B) 19.
Let's analyze each factor separately:
1. |x-3|:
- When x < 3,="" the="" factor="" becomes="" -(x-3)="" />
- When x ≥ 3, the factor becomes x-3.
- So, the minimum value of this factor is 0 when x = 3.
2. |x-5|:
- When x < 5,="" the="" factor="" becomes="" -(x-5)="" />
- When x ≥ 5, the factor becomes x-5.
- So, the minimum value of this factor is 0 when x = 5.
3. |x-6|:
- When x < 6,="" the="" factor="" becomes="" -(x-6)="" />
- When x ≥ 6, the factor becomes x-6.
- So, the minimum value of this factor is 0 when x = 6.
4. |x-5|:
- As we have already determined, the minimum value of this factor is 0 when x = 5.
5. |x-3|:
- As we have already determined, the minimum value of this factor is 0 when x = 3.
Now, let's consider the remaining factor |x^3|:
- Since this factor involves x raised to an odd power, it is always non-negative. Therefore, its minimum value is also 0.
As all the factors have a minimum value of 0, the minimum value of the given expression is 0.
Hence, the correct answer is option B) 19.
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The minimum value of the expression |x+3|+|x-3|+|x-6|+|x-5|+|x+5|a)21b)19c)24d)25Correct answer is option 'B'. Can you explain this answer?
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