Question

These problems are based on first derivative test
If $f$ be a function defined on an open interval $I$ and also if $f$ be continuous at a critical point $c$ in $I$, then point $c$ is called point of inflection, if

• A. $f^{\prime}(x)$ changes sign from positive to negative as $x$ increases through $c$
• B. $f^{\prime}(x)$ changes sign from negative to positive as $x$ increases through $c$
• C. $f^{\prime}(x)$ does not change sign as $x$ increases through $c$
• D. All the above are true

Answer 1

Answer: (C)

These problems are based on working rule for finding points of local maxima or points of local minima using only the first order derivatives. If $f$ be a function defined on an open interval I. Also, if $f$ continuous at a critical point $c$ in $I$. Then
If $f^{\prime}(x)$ does not changes sign as $x$ increases throughc, then $c$ is neither a point of local maxima nor a point of local minima. Infect, such a point is called point of inflection. (Referring to the figure used in solution 178)
Note If ' $c$ ' is a point of local maxima of $f$, then $f(c)$ is a local maximum value of $f$. Similarly, if $c$ is a point of local minima of $f$, then $f(c)$ is a local minimum value of $f$.
can be geometrically explained by the figure given below.