A curve is represented parametrically by the equations `x=f(t)=a^(In(b'))and y=g(t)=b^(-In(a^(t)))a,bgt0 and a ne 1, b ne 1"`
`Where "t in R.`
The value of `f(t)/(f'(t))cdot(f''(-t))/(f'(-t))+(f(-t))/(f'(-t))cdot(f''(t))/(f'(t))AA t in R ` is

Options

A. -2
B. 2
C. -4
D. 4

Explanation

`x=f(t)=a^(In(b))=a^(t" In "b)" (1)"`
`y=g(t)=b^(-In(a^(t)))=(b^(In a))^(-t)=(a^(In b))^(-t)=a^(-t" In b")`
`therefore" "y=g(t)=a^(In(b^(-t)))=f(-t)" (2)"`
From equations (1), and (2),
`xy=1`
`because" "xy=1`
`fg=1`
`therefore" "fg''+g'f'+g'f'+gf''=0`
`"or "fg''+g'f'+g'f'+gf''=0`
`"or "(f)/(f')(g'')/(g')+(gf'')/(g'f')=-2" (3)"`
From equation (2), g(t)=f(t)
`therefore" "g'(t)=-f'(-1)`
`"and "g''(t)=f''(-1)`
Substituting in equation (3), we get
`(f(t))/(f'(t))cdot(f''(-t))/(-f'(-t))+(f(-1))/(-f'(-t))cdot(f''(t))/(f'(t))=-2`
`(f(t))/(f'(t))cdot(f''(-1))/(f'(-t))+(f(-t))/(f'(-1))cdot(f''(t))/(f'(t))=2`