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Define a continuous function on the clos...

Define a continuous function on the closed interval [a,b]

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`lim _(x to a ^(+)) f (x ) f (a) and lim _( c to b ^(-)) f (x) =f (b)`
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Statement 1: If both functions f(t)a n dg(t) are continuous on the closed interval [1,b], differentiable on the open interval (a,b) and g^(prime)(t) is not zero on that open interval, then there exists some c in (a , b) such that (f^(prime)(c))/(g^(prime)(c))=(f(b)-f(a))/(g(b)-g(a)) Statement 2: If f(t)a n dg(t) are continuou and differentiable in [a, b], then there exists some c in (a,b) such that f^(prime)(c)=(f(b)-f(a))/(b-a)a n dg^(prime)(c)(g(b)-g(a))/(b-a) from Lagranes mean value theorem.

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Find the absolute extrema of the following functions on the given closed interval. f(x)=2cosx+sin2x,[0,pi/2]

Find the absolute extrema of the following functions on the given closed interval. f(x)=x^(3)-12x+10,[1,2]

Find the absolute extrema of the following functions on the given closed interval. f(x)=6x^(4/3)-3x^(1/3),[-1,1]

Find the absolute extrema of the following functions on the given closed interval. f(x)=3x^(4)-4x^(3),[-1,2]

Let f be a continuous function defined onto [0,1] with range [0,1]. Show that there is some c in [0,1] such that f(c)=1-c

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FULL MARKS-EXAMINATION QUESTION PAPER MARCH 2019-MATHEMATICS
  1. Define diagonal and scalar matrices?

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  2. Find a unit vector along the direction of the vector 5 hati - 3 hatj +...

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  3. Define a continuous function on the closed interval [a,b]

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  4. Consider the function f (x) = sqrtx, x ge 0. Does lim (x to 0) f (x) e...

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  5. An integer is chosen at random from the first ten positive integers.F...

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  6. It is correct to say A xxA = {(a,a) :a in A} ? Justify your answer.

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  7. A football player can kick a fotball from ground level with an initial...

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  8. Find the coefficient of x ^(3) in the expansion of (2 -3x)^(7).

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  9. Find the nearest point on the line x - 2y=5 from the origin.

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  10. If veca ,vecb, vec c are three vectors such that veca + 2 vecb + vecc ...

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  11. Examine the continuity of the following : cotx+tanx

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  12. Differentiate the following : y=sin^(-1)((1-x^(2))/(1+x^(2)))

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  13. Find (dy)/(dx) if x =a ( t - sin t), y =a (1- cos t)

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  14. Evaluate: int(x-3)sqrt(x+2)dx.

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  15. Construct a suitable domain X such that f:x to N defined by f (n) = n ...

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  16. For the given bases curve y = sin x, draw y = 1/2 sin 2x

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  17. Solve the equation sqrt(6 - 4x - x^(2)) = x + 4

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  18. State and prove any one of the Napier's formulae.

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  19. Do the limit of the function (sin (x - [x]))/(x - [x]) exist as x to 0...

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  20. Prove that for any natural number n, a ^(n) - b ^(n) is divisible by a...

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