The function f(x) = x2 + 2 +1 is
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Class 12
>>Maths
>>Continuity and Differentiability
>>Differentiability of a Function
>>The function f(x) = (x^2 - 1)|x^2 - 3x +
Question
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The function f(x)=(x
2
−1)∣x
2
−3x+2∣+cos∣x∣ is non-differentiable at
Medium
Solution
verified
Verified by Toppr
Correct option is D)
We know that function ∣x∣ is not differentiable at x=0
Therefore, ∣x
2
−3x+2∣=∣(x−1)(x−2)∣
Hence, it is not differentiable at x=1 and 2
Now, f(x)=(x
2
−1)∣x
2
−3x+2∣+cos∣x∣ is not differentiable at x=2.
For 1<x<2,f(x)=−(x
2
−1)(x
2
−3x+2)+cosx
For 2<x<3,f(x)=(x
2
−1)(x
2
−3x+2)+cosx
Lf
′
(x)=−(x
2
−1)(2x−3)−2x(x
2
−3x+2)−sinx
Lf
′
(2)=−3sin2
Rf
′
(x)=(x
2
−1)(2x−2)+2x(x
2
−3x+2)−sinx
Rf
′
(2)=(4−1)(4−3)+0−sin2=3−sin2
Hence, Lf
′
(2)
=Rf
′
(2).
So, f(x) is not differentiable at x=2.
Ans:
The function f(x)=(x
2
−1)∣x
2
−3x+2∣+cos∣x∣ is non-differentiable at
Medium
Solution
verified
Verified by Toppr
Correct option is D)
We know that function ∣x∣ is not differentiable at x=0
Therefore, ∣x
2
−3x+2∣=∣(x−1)(x−2)∣
Hence, it is not differentiable at x=1 and 2
Now, f(x)=(x
2
−1)∣x
2
−3x+2∣+cos∣x∣ is not differentiable at x=2.
For 1<x<2,f(x)=−(x
2
−1)(x
2
−3x+2)+cosx
For 2<x<3,f(x)=(x
2
−1)(x
2
−3x+2)+cosx
Lf
′
(x)=−(x
2
−1)(2x−3)−2x(x
2
−3x+2)−sinx
Lf
′
(2)=−3sin2
Rf
′
(x)=(x
2
−1)(2x−2)+2x(x
2
−3x+2)−sinx
Rf
′
(2)=(4−1)(4−3)+0−sin2=3−sin2
Hence, Lf
′
(2)
=Rf
′
(2).
So, f(x) is not differentiable at x=2.