Important Questions of Continuity and Differentiability Mathematics | Zigya

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301.

limx0loge1 +x3x - 1 is equal to

  • loge3

  • 0

  • log3e

  • 1


302.

If f(x) = x, if x is irrational0, if x is rational, then f is

  • continuous everywhere

  • discontinuous everywhere

  • continuous only at x = 0

  • continuous at all rational numbers


303.

If x + y = tan-1y and d2ydx2 = fydydx, then f(y) is equal to

  • - 2y3

  • 2y3

  • 1y

  • - 1y


304.

If f(x) = 2a - x when - a < x <a3x - 2a when a  x. Then, which of the following is true ?

  • f(x) is not differentiable at x = a

  • f(x) is discontinuous at x = a

  • f(x) is continuous for all x < a

  • f(x) is differentiable for all x  a


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305.

If f(x) = cos-11132cosx - 3sinx. Then f'(0.5) is equal to

  • 0.5

  • 1

  • 0

  • - 1


306.

If f(x) is a function such that f"(a) + f'(a) = 0 and g(x) =[f(x)]2 + [f'(x)]2 and g(3) = 8, then g(8) is equal to

  • 0

  • 3

  • 5

  • 8


307.

If f(a) = f'(x) + f"(x) + f'"(x) + ... and f(0) = 1, then f(x) is equal to

  • ex/2

  • ex

  • e2x

  • e4x


308.

If f(x) = x3 and g(x) = x3 - 4x in - 2  x , then consider the statements

(i) f(x) and g(x) satisfy mean value theorem.

(ii) f(x) and g(x) both satisfy Rolle's theorem.

(iii)  Only g(x) satisfies Rolle's theorem.

Of these statements.

  • (i) and (ii) are correct

  • only (i) is correct

  • None is correct

  • (i) and (iii) are correct


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309.

If the function f(x) defined by fx = x100100 + x9999 + ... + x22 +x+1, then f'(0) is equal to

  • 100f'(0)

  • 100

  • 1

  • - 1


310.

The function represented by the following graph is

  • continuous but not differentiable at x = 1

  • differentiable but not continuous at x = 1

  • continuous and differentiable at x = 1

  • neither continuous nor differentiable at x = 1


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