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  • The greatest integer function defined by    is differentiable at which point.</

The greatest integer function defined by f(x)=[x+2]   is differentiable at which point.

 

Option: 1

\frac{-1}{h}


Option: 2

\frac{1}{h}


Option: 3

not differentiable at any point


Option: 4

\frac{1}{2}


Answers (1)

best_answer

The greatest integer function f(x)=[x+2] where [x]  denotes the greatest integer less than or equal to x, is not differentiable at any point.

 

To see why, let's consider a specific point, say x=3 . At this point, the function value is  f(3)=[3+2]=[5]=5

To find the derivative at x=3 , calculate the limit of the difference quotient as h approaches

f^{\prime}(3)=\lim _{h \rightarrow 0} \frac{f(3+h)-f(3)]}{h}

Now, let's examine the behavior of the function around x=3 by considering values of h close to 0.

Now, let's examine the behavior of the function around by considering values of h close to 0.

for  h> 0

When h approaches 0 from the positive side (h \rightarrow 0+), f(3+h)

will always be equal to  [5+h]=5  because adding any positive value to 5  will not change the greatest integer part.

for  h> 0

When h approaches 0  from the negative side(h \rightarrow 0-), f(3+h) will be equal to  [5+h]=4

because subtracting any positive value from 5  will change the greatest integer part to 4 

Considering both cases,  rewrite the difference quotient as:


\begin{aligned} & f^{\prime}(3)=\frac{\lim (h \rightarrow 0+)[5-5]}{h}+\frac{\lim (h \rightarrow 0-)[4-5]}{h} \\ & =\frac{0}{h}+\frac{-1}{h} \\ & =\frac{-1}{h} \end{aligned}

The limit of \frac{-1}{h} as h approaches 0 does not exist, indicating that the derivative does not exist at   x=0.

This behavior holds for all points in the domain of the greatest integer function. The function experiences jumps at integer points, resulting in discontinuities and non-differentiability at those points.

Therefore, the greatest integer function f(x)=[x+2] is not differentiable at any point in its domain.


.

 

 

Posted by

Ritika Harsh

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