Check if the Mean Value Theorem can be applied to f(x)=4arccos(x) on [1,1]. If so, find all values c in (-1, 1) guaranteed by the Mean Value Theorem. If the Mean Value Theorem does not apply, then justify that conclusion.

Jack Ingram 2022-10-16 Answered
Check if the Mean Value Theorem can be applied to f ( x ) = 4 arccos ( x ) on [-1,1]
If so, find all values c in(-1,1) guaranteed by the mean Value theorem. If the Mean Value theorem does not apply, then justify that conclusion
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Answers (1)

Carly Yang
Answered 2022-10-17 Author has 19 answers
f ( x ) = 4 arccos x f ( x ) = 4 ( 1 1 x 2 ) f ( x ) = 4 1 x 2 f ( c ) = 4 1 c 2
Applying Mean Value Theorem:
f ( c ) = f ( b ) f ( a ) b a f ( c ) = 4 arccos 1 4 arccos ( 1 ) 1 ( 1 ) f ( c ) = 0 4 ( π ) 2 f ( c ) = 2 π
Now
4 1 c 2 = 2 π 2 1 c 2 = π 1 c 2 = 2 π
square on both sides.
1 c 2 = 4 π 2 c 2 = 1 4 π 2 c = ± 1 4 π 2
which gives us c=0.77 and c=-0.77
\So, Mean Value Theorem can be applied.
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