The maximum and the minimum value of the function \( f(x)=-\left|\,|x+3|-5\,\right|+2 \) are :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( Max\,[f(x)]=2;\ Min\,[f(x)] \) doesn't exist
Since a modulus is never negative,
\( \displaystyle \left|\,|x+3|-5\,\right|\ge0 \)
\( \displaystyle f(x)\le2 \)
Equality holds when \( |x+3|=5 \), that is at x = 2 or x = −8. So the maximum value is 2.
As x moves far from −3 in either direction, \( |x+3| \) grows without limit, so \( \left|\,|x+3|-5\,\right|\to\infty \) and \( f(x)\to-\infty \). The function takes no smallest value, so the minimum does not exist.