Question

If $f$ be a function defined on an open interval $I$. Suppose $c\in I$ be any point. If $f$ has a local maxima or a local minima at $x=c$, then
Statement I$f^{\prime }(c)=0$.
Statement II $f$ is not differentiable at c.

• A. Only statement I is true
• B. Only statement II is true
• C. Both the statements I and II are true
• D. Either of the statements I or II is true

Answer 1

Answer: (D)

From the previous two solutions, geometrically we can say, if $x=c$ is a point of local maxima of $f$, then the graph of $f$ around ' $c$ ' will be as shown in Fig (a). Note that function $f$ is increasing (i.e., $f^{\prime }(x)>0)$ in the interval $(c-h,c)$ and decreasing (i.e., $f^{\prime }(x)<0)$ in the interval $(c,c+h)$.
This suggests that $f^{\prime }$ (c) must be zero.

Similarly, if ' $c$ ' is a point of local minima of $f$, then the graph of $f$ around ' $c$ ' will be as shown in Fig (b). Here, $f$ is decreasing (i.e., $f^{\prime }(x)<0)$ in the interval $(c-h,c)$ and increasing (i.e., $f^{\prime }(x)>0)$ in the interval $(c,c+h)$. This again suggest that $f^{\prime }(c)$ must be zero.
The above discussion lead us to the following result
Let $f$ be a function defined on an open interval $I$. Suppose $c\in I$ be any point. If $f$ has a local maxima or a local minima at $x=c$, then either $f^{\prime }(c)=0$ or $f$ is not differentiable at $c$.