Let a∈ Z and [t] be the greatest integer t. Then the number of points, where the function f(x)=[a+13 sin x], x ∈ (0, π) is not differentiable, is ____
Correct Answer: 25
Solution and Explanation
Given Function:
\(f(x) = \lfloor a + 13 \sin x \rfloor\) for \(x \in (0, \pi)\), where \(a\) is an integer and \(\lfloor t \rfloor\) is the greatest integer less than or equal to \(t\).
Since \(0 < x < \pi\), we have \(0 < \sin x \leq 1\), and thus \(0 < 13 \sin x \leq 13\).
The greatest integer function is discontinuous at integer values of \(x\). So, \(\lfloor 13 \sin x \rfloor\) will be discontinuous when \(13 \sin x\) takes integer values from 1 to 13.
For each integer \(k\) from 1 to 12, the equation \(13 \sin x = k\) has two solutions in the interval \((0, \pi)\).
The equation \(13 \sin x = 13\) has only one solution in the interval \((0, \pi)\), which is \(x = \frac{\pi}{2}\).
Therefore, the number of points of discontinuity is:
\[ 2 \times 12 + 1 = 25 \]
The function \(f(x)\) is not differentiable at 25 points in \((0, \pi)\).
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