Practicing Success
Let y = f(x) be the equation of a parabola which is touched by the line y = x at the point where x = 1. Then, |
f'(0) = f'(1) f'(1) = -1 f(0) + f'(0) + f''(0) = 1 2f(0) = 1 - f'(0) |
2f(0) = 1 - f'(0) |
Let $y=a x^2+b x+c$ be the given parabola. Then, $f(x)=a x^2+b x+c$ Clearly, $\frac{d y}{d x}=2 a x+b$ It is given that y = x touches the parabola at x = 1. ∴ $\left(\frac{d y}{d x}\right)_{x=1}$ = (Slope of the line y = x) $\Rightarrow 2 a+b=1$ .......(i) Putting x = 1 in y = x, we get y = 1 So, the line y = x touches the parabola $y=a x^2+b x+c$ at (1, 1). $a+b+c=1$ .......(ii) Now, $f(x) =a x^2+b x+c \Rightarrow f'(x)=2 a x+b$ and $f"(x)=2 a$ ∴ $f(0) =c, f'(0)=b, f"(0)=2 a$ and $f'(1)=2 a+b$ From (ii), we have $a+b+c=1$ $\Rightarrow 2 a+2 b+2 c=2$ $\Rightarrow 2 a+b+(b+2 c)=2$ $\Rightarrow 1+(b+2 c)=2$ [∵ 2a + b = 1 from (i)] $\Rightarrow b+2 c=1 \Rightarrow 2 c=1-b \Rightarrow 2 f(0)=1-f'(0)$ Also, $f'(1)=2 a+b=1$ [Using (i)] |