Question

Find f (x )if f apostrophe apostrophe (x )equals x to the power of negative 2 end exponent comma space x greater than 0 comma space f (1 )equals 0 and f (2 )equals 0.

Answer 1

Explanation:

Given
`g'(x) = g^(-2), x> 0\ g(1) = 0 \ and \ g(2) = 0`

To get g(x), we will have to integrate g'(x)

`g(x) = \int\limits {x^(-2)} \, dx\\ g(x) = (x^(-2+1))/(-2+1) + C\\ g(x) = -x^(-1) + C\\g(x) = -x^(-1) + C\\`

If g(1) = 0, this means at x = 1, g(x) = 0

0 = -1⁻¹ + C

C= 1

Substitute C = 1 into the function

g(x) = -x⁻¹ + 1

If g(2) = 0, this means at x = 2, g(x) = 0

0 = -2⁻¹ + C

C= 2⁻¹

C = 1/2

Substitute C = 2 into the function

g(x) = -x⁻¹ + 1/2