Answer :
Final answer:
To find the largest value f(10) could take, we use the Mean Value Theorem and find that f(10) = 56.
Explanation:
To find the largest value f(10) could take, we can use the Mean Value Theorem. According to the theorem, if a function is continuous and differentiable on an interval, then there exists a point within the interval where the derivative of the function is equal to the average rate of change of the function over that interval. In this case, since the derivative of f is always less than or equal to 10, we can determine that the maximum rate of change of f on the interval [5,10] is 10.
Given that f(5) = 6, the average rate of change of f over the interval [5,10] is (f(10) - f(5))/(10 - 5) = (f(10) - 6)/5 = 10.
Therefore, we can solve for f(10) by setting the average rate of change equal to 10 and solving the equation: (f(10) - 6)/5 = 10. Solving this equation yields f(10) = 56.
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