The function f(x) = |x 2| + |2.5 x| + |3.6 x|, attains a minimum

Question

The function  f(x) = |x – 2| + |2.5 – x| + |3.6 – x| , attains a minimum at

A. x = -1
B. x = 0 
C. x = 2.3
D. x = 2.5
E. x = 2.7

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walterwhite756 · Answer

Minimum value of |x| can be zero. Also, to minimize f(x), x should be in between 2 to 3.6 otherwise f(x) will never by minimized.

Simply put x = 2.3, f(x) = 1.8,   x = 2.5, f(x) = 1.6   and x = 2.7, f(x) = 1.8.

So  D is the answer .

8.toastcrunch · Answer

f(2.5) = |2.5 - 2| + |2.5 - 2.5| + |3.6 - 2.5| = 0.5 + 0 + 1.1 = 1.6

Comparing the results, we can see that f(2.5) = 1.6 is indeed smaller than f(2.7) = 1.8

Answer is D

Adidas1424 · Answer

walterwhite756

Could you explain as to why x should be between 2 and 3.6 for f(x) to be minimized? Also if the question asked us to maximize f(x), where should the value of x lie?

walterwhite756 · Answer

Hi Adidas1424 , please find my response as follows: x has to be between 2 and 3.6 for f(x) to be minimized because of nature of f(x). See, f(x) = |x – 2| + |2.5 – x| + |3.6 – x| Now, recall that modulus function represents distance between its parameters. So, |x – 2| is nothing but distance of x from 2 on number line. Now once we get that clear, we can rephrase the question to minimize the distance of x from 2, 2.5 and 3.6 on number line. That's why x has to lie between 2 and 3.6. Now based on same understanding as above, we are asked to maximize the distance of x from 2, 2.5 and 3.6 on number line. In this question, x can lie to any of the infinities (+∞, -∞). Suppose if we were given that range of x is from -20 to +20 only. Then in that case, we need to calculate f(20) and f(-20) and maximum of both would have been the answer. Hope it helped.

Raghunandan19 · Answer

This is a question about finding the minimum value of a function composed of the sum of absolute values.

The function is given by:

f(x)=∣x−2∣+∣2.5−x∣+∣3.6−x∣
💡 Principle of Absolute Value Sums
For a function that is the sum of absolute value expressions, f(x)=∣x−a∣+∣x−b∣+∣x−c∣....

Odd Number of Terms: If there is an odd number of absolute value terms, the function attains its minimum value at the median of the critical points.

Even Number of Terms: If there is an even number of absolute value terms, the function attains its minimum value for any x between the two middle critical points (inclusive).

So find the critical points for this equation, which is 2, 2.5 and 3.6. Median is 2.5.

So minimum value would be 2.5

(Substitute and verify if needed)

The function f(x) = |x 2| + |2.5 x| + |3.6 x|, attains a minimum

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The function f(x) = |x 2| + |2.5 x| + |3.6 x|, attains a minimum

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