Q)write the values of a and b for which:

f(x)=[x](sinax)^b is

a)discontinuous at all integers

b)continuous everywhere

c)continuous everywhere but non-differentiable at all integers

d)non- differentiable at all odd integers

PLz solve this with details as I am not getting to understand this type of problems.also menion the calculus book in which this type of high level problems are there.

PLZ PLZ SIR HELP ME .

THANK YOU

To tackle the problem of determining the values of \( a \) and \( b \) for the function \( f(x) = [x](\sin(ax)) \), where \( [x] \) denotes the greatest integer function (or floor function), we need to analyze the continuity and differentiability of the function at integer points. Let's break this down step by step.

Understanding the Function

The function \( f(x) = [x](\sin(ax)) \) combines two components: the floor function \( [x] \) and the sine function scaled by \( a \). The floor function \( [x] \) is discontinuous at every integer \( n \) because it jumps from \( n-1 \) to \( n \) as \( x \) crosses \( n \). The sine function, however, is continuous everywhere.

Analyzing Discontinuity

To find values of \( a \) and \( b \) that make \( f(x) \) discontinuous at all integers, we need to ensure that the product \( [x](\sin(ax)) \) behaves in a way that it does not settle at any integer value as \( x \) approaches an integer. This can happen if \( \sin(ax) \) oscillates between values that do not allow \( f(x) \) to stabilize at the integer values of \( [x] \).

• Case a: Discontinuous at all integers

  For \( f(x) \) to be discontinuous at all integers, we can set \( a \) to be a non-integer multiple of \( \pi \). This way, \( \sin(ax) \) will not equal zero at integer points, causing \( f(x) \) to jump discontinuously. Therefore, we can choose \( a = \frac{\pi}{2} \) and \( b \) can be any value.

Ensuring Continuity

• Case b: Continuous everywhere

  For \( f(x) \) to be continuous everywhere, we need \( b = 0 \). This means that \( f(x) \) will be zero at all integers, which allows for continuity since \( \sin(ax) \) will oscillate but not affect the overall continuity of the product with \( [x] \).

Non-Differentiability at Integers

• Case c: Continuous everywhere but non-differentiable at all integers

  To achieve continuity while ensuring non-differentiability at integers, we can set \( a \) to be a rational multiple of \( \pi \) (like \( a = \pi \)) and \( b = 0 \). This keeps \( f(x) \) continuous but the floor function's jump at integers makes it non-differentiable.

• Case d: Non-differentiable at all odd integers

  For \( f(x) \) to be non-differentiable specifically at odd integers, we can set \( a \) to be an odd multiple of \( \frac{\pi}{2} \) (like \( a = \frac{3\pi}{2} \)) and \( b = 0 \). This ensures that the sine function behaves in a way that causes non-differentiability at those specific points.

Recommended Resources

For further reading and practice on similar high-level problems, I recommend looking into "Calculus: Early Transcendentals" by James Stewart or "Advanced Calculus" by Patrick M. Fitzpatrick. These texts cover a range of topics, including continuity and differentiability, with challenging problems that will help deepen your understanding.

In summary, the values of \( a \) and \( b \) can be set as follows:

• Discontinuous at all integers: \( a = \frac{\pi}{2}, b \) can be any value.
• Continuous everywhere: \( a \) can be any value, \( b = 0 \).
• Continuous everywhere but non-differentiable at all integers: \( a = \pi, b = 0 \).
• Non-differentiable at all odd integers: \( a = \frac{3\pi}{2}, b = 0 \).

Feel free to ask if you have more questions or need further clarification on any of these points!