Question

Match the following lists. Let $f\left( x \right)$ be any function
List-I
A) $f'\left( a \right) = 0$ and $f''\left( a \right) < 0$ then
B) $f'\left( a \right) = 0$ and $f''\left( a \right) > 0$ then
C) $f'\left( a \right) \ne 0$ then
D) $f'\left( a \right) > 0$

List-II
A) $f\left( x \right)$ is increasing at $x = a$
B) $f\left( x \right)$ has maximum value at $x = a$
C) $f\left( x \right)$ has neither maximum nor minimum
D) $f\left( x \right)$ has minimum value at $x = a$
E) $f\left( x \right)$ is decreasing at $x = a$

A) $A - 4,B - 2,C - 3,D - 5$
B) $A - 2,B - 4,C - 3,D - 1$
C) $A - 2,B - 4,C - 3,D - 5$
D) $A - 2,B - 4,C - 5,D - 1$

Answer 1

We can take each of the conditions in list-I. Then we can find its corresponding statement in list-II. Then we can check with the given option to find the correct option.

Complete step by step solution:
We can take the conditions one by one from list-I.
Consider condition A. $f'\left( a \right) = 0$ and $f''\left( a \right) < 0$ then,
We know that by ${2^{nd}}$ derivative test, we can say that as ${1^{st}}$ derivative at a is zero, it is a critical point and as ${2^{nd}}$ derivative at a is negative, the function has maximum value at a.
So, we can match A to statement 2 in list-II …. (1)
Now consider condition B, $f'\left( a \right) = 0$ and $f''\left( a \right) > 0$ then,
We know that by ${2^{nd}}$ derivative test, we can say that as ${1^{st}}$ derivative at a is zero, it is a critical point and as ${2^{nd}}$ derivative at a is positive, the function has minimum value at a.
So, we can match B to statement 4 in list-II …. (2)
Now consider statement C, $f'\left( a \right) \ne 0$ then.
We know that a function is maximum or minimum at critical points. As $f'\left( a \right) \ne 0$, a is not a critical point. So, we can say that the function has neither maximum nor minimum at a.
So, we can match C to statement 3 in list-II …. (3)
Now consider statement D, $f'\left( a \right) > 0$
We know that ${1^{st}}$ derivative of a function at a particular point will give the slope of the curve of the function. We are given that the derivative at a is positive. So, the function has positive slope at a. Therefore, we can say that the function is increasing at $x = a$.
So, we can match D to statement 1 in list-II …. (4)
From results (1), (2), (3) and (4), we can write,

$A - 2,B - 4,C - 3,D - 1$
So, the correct answer is option D $A - 2,B - 4,C - 3,D - 1$.

Note:
We must know that the critical point of a function is the point at which the slope of the graph will be equal to zero. Graphically, at this point the function has slope parallel to the x axis. Critical point is found by taking the ${1^{st}}$ derivative of the function and equating it to zero. Then the critical points are given by solving for the variable. We must understand that in ${2^{nd}}$ derivative test, a function has maximum when ${2^{nd}}$ derivative is negative and has minimum when ${2^{nd}}$ derivative is positive. We must not interchange the order. We must match all the statements from list-I to list -II to avoid errors.