If f (x) = 2x and g (x) = \(\frac{x^2}{2}\) + 1, then’which of the following can be a discontinuous function
  • f(x) + g(x)
  • f(x) – g(x)
  • f(x).g(x)
  • \(\frac{g(x)}{f(x)}\)
The function f(x) = \(\frac{4-x^2}{4x-x^3}\) is
  • discontinuous at only one point at x = 0
  • discontinuous at exactly two points
  • discontinuous at exactly three points
  • None of these
The set of points where the function f given by f (x) =| 2x – 1| sin x is differentiable is
  • R
  • R = {\(\frac{1}{2}\)}
  • (0, ∞)
  • None of these
The function f(x) = cot x is discontinuous on the set
  • {x = nπ, n ∈ Z}
  • {x = 2nπ, n ∈ Z}
  • {x = (2n + 1) \(\frac{π}{2}\) n ∈ Z}
  • {x – \(\frac{nπ}{2}\) n ∈ Z}
The function f(x) = e is
  • continuous everywhere but not differentiable at x = 0
  • continuous and differentiable everywhere
  • not continuous at x = 0
  • None of these
If f(x) = x² sin\(\frac{1}{x}\), where x ≠ 0, then the value of the function f(x) at x = 0, so that the function is continuous at x = 0 is
  • 0
  • -1
  • 1
  • None of these
If f(x) = is continuous at x = \(\frac{π}{2}\), then
  • m = 1, n = 0
  • m = \(\frac{nπ}{2}\) + 1
  • n = \(\frac{mπ}{2}\)
  • m = n = \(\frac{π}{2}\)
If y = log(\(\frac{1-x^2}{1+x^2}\)), then \(\frac{dy}{dx}\) is equal to
  • \(\frac{4x^3}{1-x^4}\)
  • \(\frac{-4x}{1-x^4}\)
  • \(\frac{1}{4-x^4}\)
  • \(\frac{-4x^3}{1-x^4}\)
Let f(x) = |sin x| Then
  • f is everywhere differentiable
  • f is everywhere continuous but not differentiable at x = nπ, n ∈ Z
  • f is everywhere continuous but no differentiable at x = (2n + 1) \(\frac{π}{2}\) n ∈ Z
  • None of these
If y = \(\sqrt{sin x+y}\) then \(\frac{dy}{dx}\) is equal to
  • \(\frac{cosx}{2y-1}\)
  • \(\frac{cosx}{1-2y}\)
  • \(\frac{sinx}{1-xy}\)
  • \(\frac{sinx}{2y-1}\)
The derivative of cos(2x² – 1) w.r.t cos x is
  • 2
  • \(\frac{-1}{2\sqrt{1-x^2}}\)
  • \(\frac{2}{x}\)
  • 1 – x²
The value of c in Rolle’s theorem for the function f(x) = x³ – 3x in the interval [o, √3] is
  • 1
  • -1
  • \(\frac{3}{2}\)
  • \(\frac{1}{3}\)
For the function f(x) = x + \(\frac{1}{x}\), x ∈ [1, 3] the value of c for mean value theorem is
  • 1
  • √3
  • 2
  • None of these
Let f be defined on [-5, 5] as
  • continuous at every x except x = 0
  • discontinuous at everyx except x = 0
  • continuous everywhere
  • discontinuous everywhere
Let function f (x) =
  • continuous at x = 1
  • differentiable at x = 1
  • continuous at x = -3
  • All of these
If f(x) = \(\frac{\sqrt{4+x}-2}{x}\) x ≠ 0 be continuous at x = 0, then f(o) =
  • \(\frac{1}{2}\)
  • \(\frac{1}{4}\)
  • 2
  • \(\frac{3}{2}\)
let f(2) = 4 then f”(2) = 4 then \(_{x→2}^{lim}\) \(\frac{xf(2)-2f(x)}{x-2}\) is given by
  • 2
  • -2
  • -4
  • 3
It is given that f'(a) exists, then \(_{x→2}^{lim}\) [/latex] \(\frac{xf(a)-af(x)}{(x-a)}\) is equal to
  • f – af'
  • f'(a)
  • -f’(a)
  • f (a) + af'(a)
If f(x) = \(\sqrt{25-x^2}\), then \(_{x→2}^{lim}\)\(\frac{f(x)-f(1)}{x-1}\) is equal to
  • \(\frac{1}{24}\)
  • \(\frac{1}{5}\)
  • –\(\sqrt{24}\)
  • \(\frac{1}{\sqrt{24}}\)
If y = ax² + b, then \(\frac{dy}{dx}\) at x = 2 is equal to ax
  • 4a
  • 3a
  • 2a
  • None of these
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