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

Step-by-step 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)

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

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