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For the function f(x)=(x^(100))/(100)+(...

For the function
`f(x)=(x^(100))/(100)+(x^(99))/(99)+...+(x^2)/2+x+1`
. Prove that `f^(prime)(1)=100f^(prime)(0)`.

Text Solution

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`f(x)=(x^(100))/(100)+(x^(99))/(99)+.....+(x^(2))/(2)+x+1`
`f(x) =(d)/(dx)[(x^(100)/(100)+(x^(99))/(99)+.....+(x^(2))/(2)+x+1]`
`=(1)/(100)(d)/(dx)x^(100)+(1)/(99)(d)/(dx)x^(99)`
`+......+(1)/(2)(d)/(dx)x^(2)+(d)/(dx)x+(d)/(dx)`
`=(1)/(100).100x^(99)+(1)/(99).99x^(98)`
`+....+(1)/(2)2x+1+0`
`=x^(99)+x^(98)+.....+x+1`
`rArr f(1)=1+1+.....+1+1` (upto 100 terms)=100
and `f(0)=0+0.....+0+1=1`
`therefore f(1)=100:f(1)` Hence proved.
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