Let the function be not differentiable at the two points and . Then the distance of the point from the line 12x + 5y +10 = 0 is equal to : [2025]

Question

Let the function $f (x) = (x^{2} - 1) | x^{2} - a x + 2 | + \cos | x |$ be not differentiable at the two points $x = α = 2$ and $x = β$ . Then the distance of the point $(α, β)$ from the line 12x + 5y +10 = 0 is equal to : [2025]

Smart Achievers · Accepted Answer

(*)

We have,

$f (x) = (x^{2} - 1) | x^{2} - a x + 2 | + \cos | x |$

Now, cos |x| is always differentiable

So, we will check for $| x^{2} - a x + 2 |$ and it is not differentiable at its roots.

It is given that $x = α = 2$

$\Rightarrow {(2)}^{2} - a (2) + 2 = 0 \Rightarrow 4 - 2 a + 2 = 0 \Rightarrow a = 3$

The other root of $x^{2} - 3 x + 2 is x = 1$ .

Note: There is error in question, f(x) is differentiable at x = 1.

Solution

Q.
Let the function $f (x) = (x^{2} - 1) | x^{2} - a x + 2 | + \cos | x |$ be not differentiable at the two points $x = α = 2$ and $x = β$ . Then the distance of the point $(α, β)$ from the line 12x + 5y +10 = 0 is equal to : [2025]

1 4

2 3

3 2

4 5

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let the function f(x)=(x2–1)|x2–ax+2|+cos|x| be not differentiable at the two points x=α=2 and x=β. Then the distance of the point (α, β) from the line 12x + 5y +10 = 0 is equal to : [2025]

1 4

2 3

3 2

4 5

Want Us to Email you About Special Offers & Updates?

Q.
Let the function $f (x) = (x^{2} - 1) | x^{2} - a x + 2 | + \cos | x |$ be not differentiable at the two points $x = α = 2$ and $x = β$ . Then the distance of the point $(α, β)$ from the line 12x + 5y +10 = 0 is equal to : [2025]