Question

Left hand derivative and right hand derivative of a function $f\left( x \right)$ at a point $x=a$ are defined as $f'\left( {{a}^{-}} \right)=\displaystyle \lim_{h \to {{0}^{+}}}\dfrac{f\left( a \right)-f\left( a-h \right)}{h}=\displaystyle \lim_{h \to {{0}^{-}}}\dfrac{f\left( a \right)-f\left( a-h \right)}{h}=\displaystyle \lim_{x \to {{a}^{+}}}\dfrac{f\left( a \right)-f\left( x \right)}{a-x}$ respectively. Let $f$ be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function.
If $f$ is odd, which of the following is Left-hand derivative of $f$ at $x=a$
a. $\displaystyle \lim_{h \to {{0}^{-}}}\dfrac{f\left( a-h \right)-f\left( a \right)}{-h}$
b. $\displaystyle \lim_{h \to {{0}^{-}}}\dfrac{f\left( a-h \right)-f\left( a \right)}{h}$
c. $\displaystyle \lim_{h \to {{0}^{+}}}\dfrac{f\left( a \right)+-f\left( a-h \right)}{-h}$
d. $\displaystyle \lim_{h \to {{0}^{-}}}\dfrac{f\left( -a \right)-f\left( -a-h \right)}{-h}$

Answer 1

In this type of question we use the concept of even function and odd function. If a function $f\left( x \right)$ is an odd function then $f\left( -x \right)=-f\left( x \right)$ and if $f\left( x \right)$ is an even function then $f\left( -x \right)=f\left( x \right)$. Also we have to consider the concept of $f'\left( x \right)$ that is differentiation of $f\left( x \right)$. If the differentiation of $f\left( x \right)$ is odd then $f\left( x \right)$is an even function and vice versa.

Complete step by step solution:
Now, we have to find the left-hand derivative of $f\left( x \right)$ if $f\left( x \right)$ is an odd function.
We have given that,
$\Rightarrow f'\left( {{a}^{-}} \right)=\displaystyle \lim_{h \to {{0}^{+}}}\dfrac{f\left( a \right)-f\left( a-h \right)}{h}=\displaystyle \lim_{h \to {{0}^{-}}}\dfrac{f\left( a \right)-f\left( a-h \right)}{h}=\displaystyle \lim_{x \to {{a}^{+}}}\dfrac{f\left( a \right)-f\left( x \right)}{a-x}$
We know that differentiation of an odd function is even and vice versa.
The left hand derivative of $f\left( x \right)$ at $x=a$ is given by $\displaystyle \lim_{h \to {{0}^{-}}}\dfrac{f\left( a \right)-f\left( a-h \right)}{h}$
Since, $f\left( x \right)$ is an odd function then we can say that the derivative of $f\left( x \right)$ is an even function and hence, $f'\left( x \right)$ is an even function.
Then the left-hand derivative of $f\left( x \right)$ at $x=a$ is $\displaystyle \lim_{h \to {{0}^{-}}}\dfrac{f\left( a-h \right)-f\left( a \right)}{-h}$

Thus we can say that option (a) is the correct answer.

Note:
In this type of question students always have to remember the conditions of even function and odd function. Also students have to remember about the property that if a function is odd then the differentiation of the function is an even function and vice versa.