y=f(x) is a parabola, having its axis parallel to y axis. If the line y=x touches this parabola at x=1, then find the relation between f"(1) and f'(0).

Dear Jayati

we can write general equation of parabola whose vertics is parallel to y axis

(y-b) = c(x-a)²

y = c(x-a)² + b

so f(x) = c(x-a)² + b

now y=x is tangent to parabola

f'(x) = 2c(x-a)

f''(x) = 2c

f'(1)=1 = 2c(1-a) 

 1 = 2c -2ca   ............1

f'(0) = -2ca

f''(1) =2c

put these value in equation 1

 1 = f''(1) +f'(0)

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