JAMIA 2025 — Mathematics PYQ
JAMIA | Mathematics | 2025The function f(x)=|x| is:
Choose the correct answer:
- A.
Differentiable everywhere
- B.
Continous but not differentiable at x=0
(Correct Answer) - C.
Discontinuous
- D.
None of these
Continous but not differentiable at x=0
Explanation
The given function is continuous but not differentiable at x=0.
f(x)=∣x∣={</span><br><spanstyle="font−size:14pt;">x,</span><br><spanstyle="font−size:14pt;">−x,amp;x≥0amp;xlt;0</span><br><spanstyle="font−size:14pt;">
L.H.L at x=0:
limh→0(−h)=0
R.H.L at x=0:
limh→0(0+h)=0
⇒f(0−)=f(0+)=0
Hence, the function is continuous at x=0.
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For differentiability:
f′(0−)=L.H.D=limh→0−hf(0−h)−f(0)
=limh→0−h−(−h)−0=limh→0−hh=−1
f′(0+)=R.H.D=limh→0hf(0+h)−f(0)
=limh→0h0+h−0=limh→0hh=1
Since L.H.D = R.H.D,
the function is **not differentiable at x=0**.
Explanation
The given function is continuous but not differentiable at x=0.
f(x)=∣x∣={</span><br><spanstyle="font−size:14pt;">x,</span><br><spanstyle="font−size:14pt;">−x,amp;x≥0amp;xlt;0</span><br><spanstyle="font−size:14pt;">
L.H.L at x=0:
limh→0(−h)=0
R.H.L at x=0:
limh→0(0+h)=0
⇒f(0−)=f(0+)=0
Hence, the function is continuous at x=0.
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For differentiability:
f′(0−)=L.H.D=limh→0−hf(0−h)−f(0)
=limh→0−h−(−h)−0=limh→0−hh=−1
f′(0+)=R.H.D=limh→0hf(0+h)−f(0)
=limh→0h0+h−0=limh→0hh=1
Since L.H.D = R.H.D,
the function is **not differentiable at x=0**.