edited by
3,004 views
3 votes
3 votes

Let $f$ be a real-valued function of a real variable defined as $f(x) = x^{2}$ for $x\geq0$ and $f(x) = -x^{2}$ for  $x < 0$.Which one of the following statements is true?

  1. $f(x) \text{is discontinuous at  x = 0}$ 
  2. $f(x) \text{is continuous but not differentiable at x = 0} $
  3. $f(x) \text{is differentiable but its first derivative is not continuous at  x = 0 }$
  4. $f(x) \text{is differentiable but its first derivative is not differentiable at x = 0} $
edited by

1 Answer

Best answer
9 votes
9 votes

option D) is correct.

$f(x)=x^2 , x>=0$

      $=-x^2 , x<0$

at $x=0$ function f(x) is continuous since $LHL=RHL=f(0) = 0$

now, first derivative of f(x)

$f'(x) = 2x , x>=0$

       $=-2x , x<0$

since, $f'(x=0^+)=0$ and $f'(x=0^-)=0$ i.e both are equal,

function f(x) is diffrentiable at $x=0$

derivative of first derivative,

$f''(x) = 2 , x>=0$

      $ =-2 , x<0$

since $f''(x=0^+) = 2$ and $f''(x=0^-) = -2$ i.e both are unequal

derivative of f(x) is not diffrentiable at $x=0$

selected by

Related questions

3 votes
3 votes
1 answer
1
0 votes
0 votes
0 answers
2
bts asked Jun 25, 2018
387 views
Why is a function not differentiable at x=k when f'(x) limits to infinity? Limit can be infinite too?
1 votes
1 votes
0 answers
3
Lakshman Bhaiya asked Feb 21, 2018
581 views
The number of roots of the polynomial, $s^{7} + s^{6} + 7s^{5} + 14s^{4} + 31s^{3} + 7s^{2}+ 25s + 200$, in the open left half of the complex plane is$3$$4$$5$$6$
0 votes
0 votes
0 answers
4
Lakshman Bhaiya asked Feb 21, 2018
1,159 views
Let $f(x) = 3x^{3} - 7x^{2} + 5x + 6$. The maximum value of $f(x)$ over the the interval $[ 0 , 2 ]$ is_________(Upto $1$ decimal place)