Question
If $$f''\left( x \right) < 0,\forall \,x\, \in \left( {a,\,b} \right),$$ then $$f'\left( x \right) = 0$$ occurs :
A.
exactly once in $$\left( {a,\,b} \right)$$
B.
at most once in $$\left( {a,\,b} \right)$$
C.
at least once in $$\left( {a,\,b} \right)$$
D.
none of these
Answer :
at most once in $$\left( {a,\,b} \right)$$
Solution :
Suppose, there are two points $${x_1}$$ and $${x_2}$$ in $$\left( {a,\,b} \right)$$ such that $$f'\left( {{x_1}} \right) = f'\left( {{x_2}} \right) = 0.$$ By Rolle's theorem applied to $$f'$$ on $$\left[ {{x_1},\,{x_2}} \right],$$ there must be a $$c\, \in \,\left( {{x_1},\,{x_2}} \right)$$ such that $$f''\left( c \right) = 0.$$ This contradicts the given condition $$f''\left( x \right) < 0,\forall \,x\, \in \left( {a,\,b} \right).$$
Hence, our assumption is wrong. Therefore, there can be at most one point in $$\left( {a,\,b} \right)$$ at which $$f'\left( x \right)$$ is zero.