Question

Let $f:[0,1] \rightarrow R$ be a twice differentiable function in $(0,1)$ such that $f(0)=3$ and $f(1)=5$. If the line $y=2 x+3$ intersects the graph of $f$ at only two distinct points in $(0,1)$, then the least number of points $x \in(0,1)$, at which $f ^{\prime \prime}( x )=0$, is ___

Answer 1

$ f^{\prime}(a)=f^{\prime}(b)=f^{\prime}(c)=2$
$ \Rightarrow f^{\prime \prime}(x) \text { is zero }$
for atleast $x_1 \in(a, b) \& x_2 \in(b, c)$