Help Please!!

[cole123]

find the function f(x) the graph passes through (1,2) and whose tangent line at the point (x,F(x)) has a slope of 1/x^3

 

[soroban]

Hello, cole123!

\(\displaystyle \text{Find the function }f(x)\,\text{ if the graph passes through (1,2)}\)

\(\displaystyle \text{and whose tangent line at the point }(x,f(x))\text{ has a slope of }\frac{1}{x^3}\)

\(\displaystyle \text{We are given: }\;f'(x) \:=\:x^{-3}\)

. . \(\displaystyle \text{Hence: }\:f(x) \;=\;-\frac{1}{2}x^{-2} + C \;=\;-\frac{1}{2x^2} + C\)


\(\displaystyle \text{We are given: }\;f(1) \,=\,2 \quad\Rightarrow\quad -\frac{1}{2(1^2)} + C \:=\:2 \quad\Rightarrow\quad C \:=\:\frac{5}{2}\)


\(\displaystyle \text{Therefore: }\;\boxed{f(x) \;=\;-\frac{1}{2x^2} + \frac{5}{2}}\)