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Find the value in the interval, which satisfies the Mean Value Theorem for the function () = 2 + +
on [, ].ff cc
Step 1: () = 2 + + [Write the function.]
Step 2: Since () is continuous in [, ] and differentiable in (, ), Mean Value Theorem is applicable.
Step 3: () = 2 + + [Find ().]
Step 4: () = 2 + + [Find ().]
Step 5: ′() = 2 + [Find ′().]
Step 6: ′() = 2 + [Find ′().]
Step 7: By Mean Value Theorem, there exists (, ) such that ′() = .
Step 8: 2 + =
Step 9: = (, )[Solve for .]
Step 10: At = in (, ), the function () satisfies the Mean Value Theorem.