Q. 23

# Which fraction exceeds its pth power by the greatest number possible?

Answer :

Given,

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 - x^{p} ------ (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:

---- (2)

[Since ]

To find the critical point, we need to equate equation (2) to zero.

1 = px^{p-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:

----- (3)

[Since ]

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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