Find the maximum and minimum values, if any, without using derivatives, of the following function, $f(x)=\left| x+2 \right|$ on R.
Hint: First try to remove the modulus sign from the given function by substituting the expression inside the modulus equal to 0. Now, let ‘n’ be the number obtained by this process. Now, if we will take the number smaller than ‘n’, we will get a positive result, as the modulus of any number is always positive except 0. If we will take any number greater than ‘n’ then also we will get a positive result. Therefore, the minimum of the given function will be 0, at (x = n). Here, in the above question, maximum value cannot be determined because we cannot choose the highest value of ‘x’.
Complete step-by-step solution - We have been given the function, $f(x)=\left| x+2 \right|$, which is defined on all real numbers. Now, substituting x+2 = 0, we get, x = -2. This implies, at x = -2 the function value is 0. Also, we know that modulus of any number is the magnitude of that number, irrespective of their sign. Therefore, modulus of any number is always positive. Now, if we substitute the values of ‘x’ less than -2 in the function then (x+2) will be negative but its modulus will be positive. Also, if we will substitute the values of ‘x’ greater than -2 in the function then (x+2) will be positive, so its modulus will be positive. Hence, we can conclude that the minimum value of the function, $f(x)$ is 0, at x = -2. Now, we can substitute any highest possible value of ‘x’ which cannot be determined or we can say that we can substitute the value of ‘x’ as infinity. Therefore, there is not any fixed maximum value of the function, $f(x)$.
Note: One may note that, there can be another method to find the range of this modulus function. The method is known as graph method. We have to draw the graph of this function in the real plane. Then we have to find that what is the least value of the y-coordinate in the graph of the function. That will be the minimum value of the function. Now, for maximum value, we have to check the highest value of the y-coordinate in the graph of the function, which in the above case will be infinity.
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