Question

Suppose that the equation $f\left(x\right)=x^2+bx+c=0$ has two distinct real roots $\alpha ,\beta$. The angle between the tangent to the curve $y=f\left(x\right)$ at the point $(\frac{\alpha +\beta }{2},f\left(\frac{\alpha +\beta }{2}\right))$ and the positive direction of the x-axis is

• A. $0^o$
• B. ${30}^o$
• C. ${60}^o$
• D. ${90}^o$

Answer 1

The correct option is A $0^o$

$f\left(x\right)=x^2+bx+c$

Slope of $y=f\left(x\right)$

$\frac{dy}{dx}=2x+b$

${\left(\frac{dy}{dx}\right)}_{\frac{\alpha +\beta }{2},f\left(\frac{\alpha +\beta }{2}\right)}=2\times \frac{\alpha +\beta }{2}+b$

from $f\left(x\right)=x^2+bx+c$ we know $\alpha +\beta =-b$

$\therefore \left(\frac{dy}{dx}\right)=-b+b=0[\tan 0°=0=\frac{dy}{dx}]$

Thus, angle between tangent and x-axis$=0°$