JEE MAIN - Mathematics (2019 - 9th April Morning Slot - No. 3)
Let ƒ(x) = 15 – |x – 10|; x $$ \in $$ R. Then the set
of all values of x, at which the function,
g(x) = ƒ(ƒ(x)) is not differentiable, is :
{10,15}
{5,10,15,20}
{10}
{5,10,15}
Explanation
ƒ(x) = 15 – |x – 10|
g(x) = ƒ(ƒ(x)) = 15 – |ƒ(x) – 10|
= 15 – |15 – |x – 10| – 10|
= 15 – |5 – |x – 10||
As this is a linear expression so it is non differentiable when value inside the modulus is zero.
So non differentiable when
x – 10 = 0 $$ \Rightarrow $$ x = 10
and 5 – |x – 10| = 0
$$ \Rightarrow $$ |x – 10| = 5
$$ \Rightarrow $$ x - 10 = $$ \pm $$ 5
$$ \Rightarrow $$ x = 5, 15
$$ \therefore $$ g(x) is not differentiable at x = 5, 10, 15.
g(x) = ƒ(ƒ(x)) = 15 – |ƒ(x) – 10|
= 15 – |15 – |x – 10| – 10|
= 15 – |5 – |x – 10||
As this is a linear expression so it is non differentiable when value inside the modulus is zero.
So non differentiable when
x – 10 = 0 $$ \Rightarrow $$ x = 10
and 5 – |x – 10| = 0
$$ \Rightarrow $$ |x – 10| = 5
$$ \Rightarrow $$ x - 10 = $$ \pm $$ 5
$$ \Rightarrow $$ x = 5, 15
$$ \therefore $$ g(x) is not differentiable at x = 5, 10, 15.
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