Find the value of `a` and `b` if `f(x) = { tt ((ax^2+1 , x le 1) , ( x^2 + ax+b, x > 1))` is differentiable at `x = 1`.
Solution:
`a=2, b=0`
Q 1765601565
Find the values of `p` and `q`, so that `f(x) = { tt( (x^2 +3x+p , text(if) x le 1) ,(qx +2, text(if) x >1) )` is differentiable at `x=1`. NCERT Exemplar
Solution:
We have,
`f(x) = { tt( (x^2 +3x+p , text(if) x le 1) ,(qx +2, text(if) x >1) )` is differentiable at `x=1`.
`=> q-2 -p =0 => p -q =-2` ......................(i)
`=> lim_(h-> 0) (qh +0)/h =q` [for existing the limit]
If `L H D f'(1) = R H D f'(1) `, then `5 =q`
`=> p-5 = -2 => p =3`
`:. p =3` and `q =5`
Differentiability At point where function is not changing
Q 1659167014
Consider the function `f(x) = tt { ( (x^2 - 5 , x <= 3) ,( sqrt(x + 13) , x > 3) )`
Consider the following statements
1. The function is discontinuous at `x = 3`.
2. The function is not differentiable at `x =0`.
Which of the above statement (s) is/are correct? NDA Paper 1 2014
Let `f(x)` be a function defined in `1 <= x < oo` by
` f(x) ={tt {(2-x, text(for) 1<= x <= 2),( 3x - x^2, text(for) x > 2))`
Consider the following statements
I. The function is continuous at every point in the interval `[1, oo)`,
II. The function is differentiable at `x = 1.5`.
Which of the above statements is/are correct? NDA Paper 1 2014
(A)
Only 1
(B)
Only 2
(C)
Both 1 and 2
(D)
Neither 1 nor 2
Solution:
Given function, `f(x) = {tt {(2-x, text(for) 1<= x <= 2),( 3x - x^2, text(for) x > 2))`
and whole function defined in `1 <= x < oo`.
I. Since, the function is polynomial, so it is continuous
as well as differentiable in its domain `[1, oo) - {2}`.
Now, we check the continuity of the function at `x = 2`.
`LHL = f(2 - 0)= lim_(h -> 0) f(2- h)`
`= lim_(h -> 0) 2- (2 -h)`
` = lim_(h -> 0) h = 0`
`RHL = f (2 + 0) = lim_(h -> 0) (2 +h)`
` = lim_(h -> 0) 3(2 + h) - (2 + h)^2`
` = 3(2 + 0) - (2 + 0)^2`
` = 6 - 4 = 2`
and `f(2) = 2 - 2 = 0`
`∵ f(2) = LHL != RHL`
So, the function is not continuous at every point in the interval
`[1, oo)` i.e., not continuous at `x = 2`.
II. We also check the differentiability of the function at