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if f(x) = integral x^(m-1) dx then f^ (m+1) (x)= 0 then what can we say about m ?
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if f(x) = integral x^(m-1) dx then f^ (m+1) (x)= 0 then what can we sa...
Solution


Given:

f(x) = integral x^(m-1) dx


To find:

f^(m 1) (x)= 0

What can we say about m?


Solution:

To find f^(m 1) (x), we need to differentiate f(x) m times.

Using the power rule of differentiation, we get:


f'(x) = x^(m-1)

f''(x) = (m-1)x^(m-2)

f'''(x) = (m-1)(m-2)x^(m-3)


We can observe that the nth derivative of f(x) is given by:


f^(n) (x) = (m-1)(m-2)...(m-n+1)x^(m-n)


For f^(m 1) (x) to be 0, we need (m-1)(m-2)...(m-m+1) = 0, i.e., (m-1)! = 0.

This is possible only when m = 1.


Conclusion:

Hence, we can say that if f(x) = integral x^(m-1) dx and f^(m 1) (x)= 0, then m = 1.
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if f(x) = integral x^(m-1) dx then f^ (m+1) (x)= 0 then what can we say about m ?
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