Which fraction exceeds its pth power by the greatest number possible?
The pth power of a number exceeds by a fraction to be the greatest.
Let us consider,
• ‘x’ be the required fraction.
• The greatest number will be y = x - xp ------ (1)
For finding the maximum/ minimum of given function, we can find it by differentiating it with x and then equating it to zero. This is because if the function y(x)has a maximum/minimum at a point c then y’(c) = 0.
Differentiating the equation (1) with respect to x:
To find the critical point, we need to equate equation (2) to zero.
1 = pxp-1
Now to check if this critical point will determine the if the number is the greatest, we need to check with second differential which needs to be negative.
Consider differentiating the equation (2) with x:
Now let us find the value of
As , so the number y is greatest at
Hence, the y is the greatest number and exceeds by a fraction
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