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

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

Tardigrade Admin · Accepted Answer

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^{'} (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.

Q. 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

$f^{'} (x)$ changes sign from positive to negative as $x$ increases through $c$

$f^{'} (x)$ changes sign from negative to positive as $x$ increases through $c$

$f^{'} (x)$ does not change sign as $x$ increases through $c$

All the above are true

Q. 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

f′(x) changes sign from positive to negative as x increases through c

f′(x) changes sign from negative to positive as x increases through c

f′(x) does not change sign as x increases through c

All the above are true

Q. 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

$f^{'} (x)$ changes sign from positive to negative as $x$ increases through $c$

$f^{'} (x)$ changes sign from negative to positive as $x$ increases through $c$

$f^{'} (x)$ does not change sign as $x$ increases through $c$