Question

Let $f$ be a continuous, differentiable and bijective function. If the tangent to $y=f\left(x\right)$ at $x=b$, then there exists at least one $c\in (a,b)$ such that

• A. $f^{\prime }\left(c\right)=0$
• B. $f^{\prime }\left(c\right)>0$
• C. $f^{\prime }\left(c\right)<0$
• D. none of these

Answer 1

The correct option is A $f^{\prime }\left(c\right)=0$

Since the same line is tangent at one point $x=a$ and normal at other point $x=b$

$\Rightarrow$ Tangent at $x=b$ will be perpendicular to tangent at $x=a$

$\Rightarrow$ Slope of tangent changes from positive to negative or negative to positive. Therefore, it takes the value zero somewhere. Thus, there exists a point $c\in (a,b)$ where $f^{\prime }\left(c\right)=0$.