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Let f be a differentiable function on [a,b ] and x belongs to[a,b]. Show that, if f'(x)=0 and f''(x)>0, then f must have a local maximum at x.

Let f :[0,1] tends to R be a function defined by f(x)=x^m (1-x)^n, where m,n belongs to N.Find the values of m and n such that the Rolle’s Theorem holds for the function f .

Find the following limit x tends to 0 (1-cosx^2/x^2 - x^2 sin x^2)

Determine the local minimum and local maximum value of the function f defined by f(x)=3-5x^3+5x^4-x^5

If the nominal interest rate is 3%, how much is P5,000 worth in 10 years in a continuous compounded account?

Prove that a strictly decreasing function is always one-one

Find the following limit

Lim x tengs to 0 1-cos x^2/x^2 sin x^2

e) Evaluate lim x infty m 1+n^ 2 + m 4+n^ 2 + m 9+n^ 2 +***+ n 2n^ 2 ]

  1. Use mathematical induction to show that n! ≥ 2(n-1) for all n ≥ 1.
  2. Use (1) and the definition of a Cauchy sequence to show that Sn = ( 1+ 1/2! + 1/3! + ⋯ 1/n!) is Cauchy sequence.
  1. Use the definition of the limit  to show that the sequence (1 + (−1)^n) is divergent.
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