Find, if it exists, a value c in the interval [1, 4] such that the instantaneous rate of change of f(x) = 12 at c is the same as the average rate of change of f over the interval [1, 4]. (If an answer does not exist, enter DNE.)

  c = 2.25

Step-by-step explanation:

f(x) is a continuous and differentiable function on the interval, so the Mean Value Theorem guarantees a value for c exists.

The average slope is ...

  m = (f(4) -f(1))/(4 -1) = (24 -12)/3 = 4

The point at which the derivative is 4 is ...

  f'(c) = 6/√c = 4

  √c = 6/4

  c = 2.25

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