First Derivative Test - Practice Questions & MCQ

Critical Points:

Extremas of a fucntion always lie on the critical points only. A critical point is a point belonging to the domain of the function such that either the function is non-differentiable at this point or the derivative of function at this point is zero.

Hence, a function $f(x)$ has critical point at $x=a$, if $f^{\prime}(a)=0$ or $f^{\prime}(a)$ is not defined.

First Derivative Test to Get Extrema

(1) At critical point $x=a$

Let $y=f(x)$ be a differentiable function and $x=$ a be a critical point of $y=f(x)$.
If $f^{\prime}(x)$ changes from positive to negative at $x=a$, then $f$ has a Local Maxima at $x=a$.

If $\mathrm{f}^{\prime}(\mathrm{x})$ changes from negative to positive at $x=a$, then $f$ has a Local Minima at $x=a$.

If $f^{\prime}(x)$ does not change any sign at $x=a$, then $f$ has neither Local maxima nor Local minima at $x=$ a.

(2) At end points of the closed interval [a, b]
$f(x)$ is defined on the interval [a, b] and if $f^{\prime}(x)<0$ for $x$ then $f(x)$ has a local maximum at $x=a$ and local minimum at $\mathrm{x}=\mathrm{b}$.

Again if $f^{\prime}(x)>0$ then $f(x)$ has a local minimum at $x=a$ and local maximum at $x=b$.

n^th derivative test

First find the value of $x$ such that $f^{\prime}(x)=0$, let at $x=a, f^{\prime}(x)=0$
Now, find f " $(\mathrm{x})$ at $\mathrm{x}=\mathrm{a}$
If $f^{\prime \prime}(a)<0$, then $f(x)$ has local maximum at $x=a$
If $f^{\prime \prime}(a)>0$, then $f(x)$ has local minimum at $x=a$
If $f^{\prime \prime}(a)=0$
Then, find $\mathrm{f}^{\prime \prime \prime}(\mathrm{x})$ at $\mathrm{x}=\mathrm{a}$
If $f^{\prime \prime \prime}(a) \neq 0$, then $f(x)$ has neither maximum nor minimum (inflection point) at $x=a$.
But, if $f^{\prime \prime \prime}(a)=0$, then find fourth derivative of $f(x)$ at $x=a$, i.e. $f^{\text {iv }}(x)$ at $x=a$.
If $f^{\text {iv }}(a)<0$, then $f(x)$ is maximum at $x=a$ and if $f^{\text {iv }}(a)>0$ then $f(x)$ is minimum at $x=a$ and so on.

SUMMARY

Maxima and Minima of Discontinuous and Non-Differentiable Functions

Discontinuous Function

If function $f(x)$ is discontinuous at $x=a$ with $f(a)$ exists finitely
In this case, we use the most common definition of minima and maxima
I.e. if $f(a)>f(a-h)$ and $f(a)>f(a+h)$, then $x=a$ is point of maxima and if $f(a)<f(a-h)$ and $f(a)<f(a$ $+h)$, then $x=a$ is point of minima.

We can also use the graph of the function to decide the point of maxima or minima for such a function.

Non-Differentiable Function
When the function $f(x)$ is continuous at $x=$ a but $f(x)$ is not differentiable at $x=a$
In this case, we can check the change in the sign of derivative in the neighbourhood of $\mathrm{x}=\mathrm{a}$
i.e. if $f^{\prime}(a-h)>0$ and $f^{\prime}(a+h)<0$, then $x=a$ is point of maxima

And if $f^{\prime}(a-h)<0$ and $f^{\prime}(a+h)>0$, then $x=a$ is point of minima.

Global Maxima and Minima

Global Maxima and Minima in [a, b]
Let $y=f(x)$ be a given function in the domain $D$ and let $[a, b] \subseteq D$. Then, global maxima and minima of a function $f(x)$ in the closed interval $[a, b]$ is the greatest and least value of $f(x)$ in $[a, b]$ respectively.
Global maxima and minima in [a, b] would always occur at critical points of the function $f(x)$ within the $[a, b]$ or at the endpoints of the interval.
First of all, we shall find out all the critical points of $f(x)$ on (a, b) i.e. points where $f^{\prime}(x)=0$ or where $f(x)$ is non-differentiable or discontinuous.
Let $c_1, c_2, \ldots c_n$ are the critical points of $f(x)$ then we shall find the value of the function for all those critical points.
Let $f\left(c_1\right), f\left(c_2\right), \ldots, f\left(c_n\right)$ be the values of the function at critical points.
Let $M_1=\max \left\{f(a), f\left(c_1\right), f\left(c_2\right), \ldots, f\left(c_n\right), f(b)\right\}$ and

$
M_2=\min \left\{f(a), f\left(c_1\right), f\left(c_2\right), \ldots, f\left(c_n\right), f(b)\right\}
$
Then $\mathrm{M}_1$ is the Global Maximum or Absolute Maximum or the Greatest value of the function and
$\mathrm{M}_2$ is the Global Minimum or Absolute Minimum or the Least value of the function.

Global Maxima and Minima in (a, b)

The method for obtaining the greatest and least values of f(x) in (a, b) is almost the same as the method used for obtaining the greatest and least values in [a, b] however with caution.

Step 1

We do not take $f(a)$ and $f(b)$ into consideration in first step
Let $M_1=\max \left\{f\left(c_1\right), f\left(c_2\right), \ldots, f\left(c_n\right)\right\}$ and

$
M_2=\min \left\{f\left(c_1\right), f\left(c_2\right), \ldots, f\left(c_n\right)\right\}
$
Step 2
If, $\quad \lim _{x \rightarrow a^{+}} f(x)>M_1$ or $\lim _{x \rightarrow b^{-}} f(x)>M_1$
Then $f(x)$ does not possess global maxima
If none of these two is true then M1 is Global Maxima

And if, $\quad \lim _{x \rightarrow a^{+}} f(x)<M_2$ or $\lim _{x \rightarrow b^{-}} f(x)<M_2$
Then $f(x)$ does not possess global minima
If none of these two conditions are satisfied, then M2 is the global minimum

For example,

This function has no global maxima and no global minima.