State Bonnet’s mean value theorem for integrals. Apply it to show that:
|3∫5 cosxdx/x|≤ 2/3
Bonnet Mean Value Theorem .
Suppose "f" is Lebesgue integrable on "[a, b]" and "g:[a,b]\\to\\R" is monotone.
i) If "g" is non-negative, decreasing and greater than or equal to "0," for "A\\in \\R," "A\\geq \\lim\\limits_{x\\to a^{+}}g(x)" there exists "C" such that "a\\leq C\\leq b" and
ii) If "g" is non-negative, increasing and greater than or equal to "0," for "B\\in \\R," "A\\geq \\lim\\limits_{x\\to b^{-}}g(x)" there exists "C" such that "a\\leq C\\leq b" and
Consider
The function "f(x)=\\cos x" is integrable on "[3, 5]."
The function "g(x)=\\dfrac{1}{x}" is non-negative, monotone decreasing on "[3, 5]."
Then by the Bonnet Mean Value Theorem, for "A\\geq \\lim\\limits_{x\\to 3^{+}}\\dfrac{1}{x}" there exists "C" such that "3\\leq C\\leq 5" and
Let "A=\\dfrac{1}{3}"
"=\\dfrac{1}{3}\\big|[\\sin x]\\big|\\begin{matrix}\n 5 \\\\\n 3\n\\end{matrix}\\leq\\dfrac{1}{3}(2)=\\dfrac{2}{3}"
Therefore
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