Advertisements
Advertisements
प्रश्न
Prove that the function f given by f(x) = |x − 1|, x ∈ R is not differentiable at x = 1.
Advertisements
उत्तर
Any function will not be differentiable if the left-hand limit and the right-hand limit are not equal.
f(x) = |x − 1|, x ∈ R
f(x) = (x − 1), if x − 1 > 0
= −(x − 1), if x − 1 < 0
At x = 1
f(1) = 1 − 1 = 0
left-side limit:
`lim_(h -> 0^-) (f(1 - h) - f(1))/ -h`
= `lim_(h -> 0^-) (1 - (1 - h) - 0)/ (- h)`
= `lim_(h -> 0^-) (+ h)/(- h)`
= −1
Right-side limit:
= `lim_(h -> 0^+) (f(1 + h) - f(1))/h`
= `lim_(h -> 0^+) ((1 + h) - 1 - 0)/ h`
= `lim_(h -> 0^+) h/h`
= 1
Left-side limit and the right-side limit are not equal.
Hence, f(x) is not differentiable at x = 1.
APPEARS IN
संबंधित प्रश्न
Differentiate the function with respect to x.
sin (x2 + 5)
Differentiate the function with respect to x.
cos x3 . sin2 (x5)
Differentiate the function with respect to x.
`cos (sqrtx)`
Differentiate the function with respect to x:
(3x2 – 9x + 5)9
Differentiate the function with respect to x:
`(5x)^(3cos 2x)`
Differentiate the function with respect to x:
`(cos^(-1) x/2)/sqrt(2x+7)`, −2 < x < 2
Find `dy/dx`, if y = 12 (1 – cos t), x = 10 (t – sin t), `-pi/2 < t < pi/2`.
If (x – a)2 + (y – b)2 = c2, for some c > 0, prove that `[1+ (dy/dx)^2]^(3/2)/((d^2y)/dx^2)` is a constant independent of a and b.
Discuss the continuity and differentiability of the
Let f(x) = x|x|, for all x ∈ R. Discuss the derivability of f(x) at x = 0
If y = tanx + secx, prove that `("d"^2y)/("d"x^2) = cosx/(1 - sinx)^2`
Differentiate `tan^-1 (sqrt(1 - x^2)/x)` with respect to`cos^-1(2xsqrt(1 - x^2))`, where `x ∈ (1/sqrt(2), 1)`
If u = `sin^-1 ((2x)/(1 + x^2))` and v = `tan^-1 ((2x)/(1 - x^2))`, then `"du"/"dv"` is ______.
|sinx| is a differentiable function for every value of x.
`sin sqrt(x) + cos^2 sqrt(x)`
sinx2 + sin2x + sin2(x2)
sinmx . cosnx
`tan^-1 (secx + tanx), - pi/2 < x < pi/2`
`tan^-1 ((3"a"^2x - x^3)/("a"^3 - 3"a"x^2)), (-1)/sqrt(3) < x/"a" < 1/sqrt(3)`
`tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2) - sqrt(1 - x^2))), -1 < x < 1, x ≠ 0`
For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.
The differential coefficient of `"tan"^-1 ((sqrt(1 + "x") - sqrt (1 - "x"))/(sqrt (1+ "x") + sqrt (1 - "x")))` is ____________.
A function is said to be continuous for x ∈ R, if ____________.
The rate of increase of bacteria in a certain culture is proportional to the number present. If it doubles in 5 hours then in 25 hours, its number would be
`d/(dx)[sin^-1(xsqrt(1 - x) - sqrt(x)sqrt(1 - x^2))]` is equal to
A particle is moving on a line, where its position S in meters is a function of time t in seconds given by S = t3 + at2 + bt + c where a, b, c are constant. It is known that at t = 1 seconds, the position of the particle is given by S = 7 m. Velocity is 7 m/s and acceleration is 12 m/s2. The values of a, b, c are ______.
Let f: R→R and f be a differentiable function such that f(x + 2y) = f(x) + 4f(y) + 2y(2x – 1) ∀ x, y ∈ R and f’(0) = 1, then f(3) + f’(3) is ______.
Let S = {t ∈ R : f(x) = |x – π| (e|x| – 1)sin |x| is not differentiable at t}. Then the set S is equal to ______.
If f(x) = `{{:(ax + b; 0 < x ≤ 1),(2x^2 - x; 1 < x < 2):}` is a differentiable function in (0, 2), then find the values of a and b.
Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.
