Mean value theorem

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  1. Consider the function f(x)=2x35x1f\left( x \right) = 2{x^3} - 5x - 1 on the interval [0, 2].
    1. Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval.
    2. Find all numbers between 0 and 2 that satisfy the conclusion of the Mean Value Theorem.
  2. If f(2)=5f\left( 2 \right) = 5 and f(x)3f'\left( x \right) \ge 3 for 2x102 \le x \le 10, what is the smallest possible value for f(10)?f\left( {10} \right)?