Advertisements
Advertisements
प्रश्न
Show that the function f(x) = |sin x + cos x| is continuous at x = π.
Advertisements
उत्तर
Given that f(x) = |sin x + cos x| at x = π
Put g(x) = sin x + cos x and h(x) = |x|
∴ h[g(x)] = h(sin x + cos x) = |sin x + cos x|
Now, g(x) = sin x + cos x is a continuous function since sin x and cos x are two continuous functions at x = π.
We know that every modulus function is a continuous function everywhere.
Hence, f(x) = |sin x + cos x| is continuous function at x = π.
APPEARS IN
संबंधित प्रश्न
Differentiate the function with respect to x.
`2sqrt(cot(x^2))`
If f(x) = |x|3, show that f"(x) exists for all real x and find it.
Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer?
If y = `[(f(x), g(x), h(x)),(l, m,n),(a,b,c)]`, prove that `dy/dx = |(f'(x), g'(x), h'(x)),(l,m, n),(a,b,c)|`.
Discuss the continuity and differentiability of the
If sin y = xsin(a + y) prove that `(dy)/(dx) = sin^2(a + y)/sin a`
`"If y" = (sec^-1 "x")^2 , "x" > 0 "show that" "x"^2 ("x"^2 - 1) (d^2"y")/(d"x"^2) + (2"x"^3 - "x") (d"y")/(d"x") - 2 = 0`
Differential coefficient of sec (tan–1x) w.r.t. x is ______.
`cos(tan sqrt(x + 1))`
(sin x)cosx
`cos^-1 ((sinx + cosx)/sqrt(2)), (-pi)/4 < x < pi/4`
`tan^-1 (sqrt((1 - cosx)/(1 + cosx))), - pi/4 < x < pi/4`
`tan^-1 (secx + tanx), - pi/2 < x < pi/2`
If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to ______.
If `"f"("x") = ("sin" ("e"^("x"-2) - 1))/("log" ("x" - 1)), "x" ne 2 and "f" ("x") = "k"` for x = 2, then value of k for which f is continuous is ____________.
A function is said to be continuous for x ∈ R, if ____________.
If `y = (x + sqrt(1 + x^2))^n`, then `(1 + x^2) (d^2y)/(dx^2) + x (dy)/(dx)` is
If `ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1, then prove that `(dy)/(dx) = - sqrt((1 - y^2)/(1 - x^2))`
Let c, k ∈ R. If f(x) = (c + 1)x2 + (1 – c2)x + 2k and f(x + y) = f(x) + f(y) – xy, for all x, y ∈ R, then the value of |2(f(1) + f(2) + f(3) + ... + f(20))| 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 ______.
If f(x) = `{{:((sin(p + 1)x + sinx)/x,",", x < 0),(q,",", x = 0),((sqrt(x + x^2) - sqrt(x))/(x^(3//2)),",", x > 0):}`
is continuous at x = 0, then the ordered pair (p, q) is equal to ______.
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 ______.
The set of all points where the function f(x) = x + |x| is differentiable, is ______.
Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.
Which expression defines the derivative of a real function \[f\] at a point \[c\] in its domain?
What is \[\frac{d}{dx}(\sin x)\]?
What is \[\frac{d}{dx}(\tan x)\]?
Which limit is the right-hand derivative at \[x=c\]?
For differentiability on a closed interval \[[a,b]\], which derivative is considered at \[b\]?
If a function \[f\] is differentiable at a point \[c\], what must be true at that point?
If \[f\] is differentiable at \[c\], which limit equals \[f'(c)\]?
For \[x\ne c\], which identity is used to prove that differentiability implies continuity?
Which conclusion establishes that \[f\] is continuous at \[x=c\]?
Which statement correctly describes the converse of “differentiability implies continuity”?
For \[f(x)=|x|\], what is the left-hand derivative at \[x=0\]?
When does a derivative exist?
