Question

Find the maximum and minimum values, if any, of the following functions:
(i) ${\text{f(x) = }}\left| {{\text{x + 2}}} \right| - 1$ (ii)${\text{g(x) = - }}\left| {{\text{x + 1}}} \right| + 3$

Answer 1

Hint: The value of the modulus can never be negative. It can be 0 or greater than 0. So put the value of modulus either equal to zero or equal to or less than zero and simplify. From there, you can find the maximum or minimum values of the modulus. To find the minimum or maximum value of the function, put the minimum or maximum value of modulus in the function.

Step-by-Step Solution:

(i ) Given function is ${\text{f(x) = }}\left| {{\text{x + 2}}} \right| - 1$.Since we know that the value of the modulus can never be negative. It can either be 0 or greater than 0. So the minimum value of$\left| {{\text{x + 2}}} \right| = 0$. This means that the function will have minimum value at this point. So we can find the minimum value of the function by putting this value in the given function. Thus, Minimum value of ${\text{f}}\left( {\text{x}} \right)$ =minimum value of $\left| {{\text{x + 2}}} \right| - 1$
$ \Rightarrow $ Minimum value of ${\text{f}}\left( {\text{x}} \right)$=$0 - 1 = - 1$
Answer-Hence, the minimum value of ${\text{f}}\left( {\text{x}} \right) = - 1$ and there is no maximum value of ${\text{f}}\left( {\text{x}} \right)$.
(ii) Given function is${\text{g(x) = - }}\left| {{\text{x + 1}}} \right| + 3$. Since we know that the value of the modulus can never be negative. It can either be 0 or greater than 0.So we can write $\left| {{\text{x + 1}}} \right| \geqslant 0$ but we need the value of${\text{ - }}\left| {{\text{x + 1}}} \right|$. So we multiply the negative sign both side in the inequality.
$ \Rightarrow {\text{ - }}\left| {{\text{x + 1}}} \right| \leqslant 0$ Which means that $0$ is the maximum value of ${\text{ - }}\left| {{\text{x + 1}}} \right|$ . So to find the maximum value of the given function, put the maximum value of modulus in the function. Thus,
Maximum value of function=maximum value of ${\text{ - }}\left| {{\text{x + 1}}} \right| + 3$
$ \Rightarrow $ Maximum value of ${\text{g}}\left( {\text{x}} \right) = 0 + 3 = 3$
Answer-Hence, the maximum value of ${\text{g}}\left( {\text{x}} \right) = 3$ and there is no minimum value of ${\text{g}}\left( {\text{x}} \right)$.

Note: When the negative sign is multiplied in any inequality, the direction of the inequality sign must be flipped or changed. We can understand this by number line. We know that the value after 0 is positive and in increasing order but the values before 0 in the number line are negative and in decreasing order. So when $\left| {{\text{x + 1}}} \right| \geqslant 0$ its value is positive and can be placed on the right side of zero on the number line but when we multiply then negative sign both side it becomes $ - \left| {{\text{x + 1}}} \right|$ , it means the number is negative and must be placed on the left side of 0 in the number line. This means it is less than zero hence the sign of inequality is changed.