Question

Let $f\left(x\right)$ be a differentiate function and $f\left(\alpha \right)=f\left(\beta \right)=0(\alpha <\beta )$, then in the interval $(\alpha ,\beta )$.

• A. $f\left(x\right)+f^{\prime }\left(x\right)=0$ has at least one root
• B. $f\left(x\right)-f^{\prime }\left(x\right)=0$ has at least one real root
• C. $f\left(x\right)\cdot f^{\prime }\left(x\right)=0$ has at least one real root
• D. None of these

Answer 1

The correct option is C $f\left(x\right)\cdot f^{\prime }\left(x\right)=0$ has at least one real root

If $f\left(x\right)$ is a differentiable function with

$f\left(\alpha \right)=f\left(\beta \right)=0(\alpha <\beta )$

then in interval $(\alpha ,\beta )$ for some part

$f^{\prime }\left(x\right)<0$

$A+$ are point

$f\left(x\right)=0$

$f\left(x\right).f^{\prime }\left(x\right)$ has at least are real root.