Question

If \[15 < x < 30\] then \[f(x) = |x - 15| + |x - 30|\] is
A. Increasing
B. Decreasing
C. Constant
D. cannot be estimated

Answer 1

Hint: Here in this question, we have to determine the function is increasing function or decreasing function or constant function. Since the given function is a mod function, by considering the interval value we determine the value for the mod function and then on simplification we obtain the function value. Then on differentiating we determine the type of function.

Complete step by step answer:
Now consider the given function \[f(x) = |x - 15| + |x - 30|\], the function is in the form of mod. The mod can either be a positive value or negative value.
In the question the interval of x is already mentioned.
Now consider the interval
\[ \Rightarrow 15 < x < 30\]-------(1)
On considering the left inequality of equation (1) we have
\[ \Rightarrow 15 < x\]
On adding -15 on both sides, we have
\[ \Rightarrow 15 - 15 < x - 15\]
On simplifying we have
\[ \Rightarrow 0 < x - 15\]
Therefore we have \[x - 15 > 0\], the value is greater than zero and it is a positive value. So we have
\[ \Rightarrow |x - 15| = x - 15\] ------(2)

On considering the right inequality of equation (1) we have
\[ \Rightarrow x < 30\]
On adding -30 on both sides, we have
\[ \Rightarrow x - 30 < 30 - 30\]
On simplifying we have
\[ \Rightarrow x - 30 < 0\]
Therefore we have \[x - 30 < 0\], the value is less than zero and it is a negative value. So we have
\[ \Rightarrow |x - 30| = - (x - 30)\] -------(3)
On substituting the equation (2) and equation (3) in the given function we have
\[ \Rightarrow f(x) = (x - 15) + ( - (x - 30))\]
On applying the sign conventions to the terms which is present in parenthesis
\[ \Rightarrow f(x) = (x - 15) + ( - x + 30)\]
On simplifying we have
\[ \Rightarrow f(x) = x - 15 - x + 30\]
On further simplifying we get
\[ \Rightarrow f(x) = 15\]
On differentiating this function we have
\[ \therefore f'(x) = 0\]
Therefore the given function is a constant function.

Hence, the correct answer is option C.

Note:To determine whether the given function is increasing function, decreasing function or constant function first we have to determine the derivative of a function.
-If a \[f'(x) > 0\] for every $x$ on some interval \[I\], then \[f(x)\] is increasing on the interval.
-If a \[f'(x) < 0\] for every $x$ on some interval \[I\], then \[f(x)\] is decreasing on the interval.
-If a \[f'(x) = 0\] for every $x$ on some interval \[I\], then \[f(x)\] is constant on the interval.