Consider the following statements for f(x) = e^-|x| ;

1. The function is continuous at x = 0.

2. The function is differentiable at x = 0.

Which of the above statements is / are correct?

Concept:

f(x) = |x| ⇒ f(x) = x if x > 0,  and f(x) =  -x, x < 0

A function f(x) is continuous at x = a, if 

A function f(x) is differentiable at x = a, if  LHD = RHD

Calculation:

Here, f(x) =  e-|x| 

So, the function is continuous at x = 0

f(x) =  e-|x| 

f'(x) = ex  for x < 0 and f'(x) =  -e-x  for x > 0

Here, LHD ≠ RHD so f(x) is not differentiable at x = 0

Hence, option (1) is correct.

Alternate MethodReferring to the graph for the function,

 f(x) = e-|x| 

 f(x) = ex for x > 0 

 f(x) = e-x for x > 0

 f(x) = 1 for x = 0

• The graph can be as,

• This will be an even function as it is symmetric about y-axis.
• We can see that the function is continuous at x = 0 as, there is no discontinuity at x = 0.
• You can see there is a sharp corner at x = 0  for the graph so this not differentiable at x = 0

• Hence, option (1) is correct.