Let `y=f(x)` be a parabola, having its axis parallel to the y-axis, which is touched by the line `y=x` at `x=1.` Then, (a)`2f(0)=1-f^(prime)(0)` (b) `f(0)+f^(prime)(0)+f(0)=1` (c)`f^(prime)(1)=1` (d) `f^(prime)(0)=f^(prime)(1)`

Options

A. 2f(0)=1-f'(0)
B. f(0)+f'(0)+f(0)=1
C. f'(1)=1
D. none of these

Answer

A

Explanation

(1) The general equation of a parabola having its axis parallel to the y-axis is
`y=ax^(2)+bx+c` (1)
This is touched by the line y=xatx=1.
Therefore, the slope of the tangent at (1,1) is 1. Also, `x=ax^(2)+bx+c` must have equal roots, i.e.,
`((dy)/(dx))_((1","1))=1and(b-1)^(2)=4ac`
`or2a+b=1and(b-1)^(2)=4ac`
Also, (1,) lies on (1). Therefore,
a+b+c=1
From 2a+b=1anda+b+c, we get a-c=0
or a=c
Substituting in a+b+c=1, we get
2c+b=1
`:." "2f(0)+f'(0)=1" "[becausef(0)=candf'(0)=b]`