Question Solved1 Answer This is a question ask for proof. Could any expert do it step by step coz I really confused about “at least one point c”. Question 1. Show that if \( f \) is continuous on a closed interval \( [a, b] \), then there exists at least one point \( c \) in \( [a, b] \) such that \[ \int_{a}^{b} f(x) d x=f(c)(b-a) . \] (Hint: Use the Mean value theorem).

FVW8GT The Asker · Calculus
This is a question ask for proof. Could any expert do it step by step coz I really confused about “at least one point c”.

Transcribed Image Text: Question 1. Show that if \( f \) is continuous on a closed interval \( [a, b] \), then there exists at least one point \( c \) in \( [a, b] \) such that \[ \int_{a}^{b} f(x) d x=f(c)(b-a) . \] (Hint: Use the Mean value theorem).
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Transcribed Image Text: Question 1. Show that if \( f \) is continuous on a closed interval \( [a, b] \), then there exists at least one point \( c \) in \( [a, b] \) such that \[ \int_{a}^{b} f(x) d x=f(c)(b-a) . \] (Hint: Use the Mean value theorem).
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Step1/2mjx-container[jax="CHTML"]{line-height: 0;}mjx-container [space="2"]{margin-left: .167em;}mjx-container [space="4"]{margin-left: .278em;}mjx-container [size="s"]{font-size: 70.7%;}mjx-assistive-mml{position: absolute !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0px 0px 0px !important; border: 0px !important; display: block !important; width: auto !important; overflow: hidden !important; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none;}mjx-math{display: inline-block; text-align: left; line-height: 0; text-indent: 0; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; border-collapse: collapse; word-wrap: normal; word-spacing: normal; white-space: nowrap; direction: ltr; padding: 1px 0;}mjx-mi{display: inline-block; text-align: left;}mjx-c{display: inline-block;}mjx-mrow{display: inline-block; text-align: left;}mjx-mo{display: inline-block; text-align: left;}mjx-msubsup{display: inline-block; text-align: left;}mjx-script{display: inline-block; padding-right: .05em; padding-left: .033em;}mjx-script > mjx-spacer{display: block;}mjx-mn{display: inline-block; text-align: left;}mjx-msup{display: inline-block; text-align: left;}mjx-c::before{display: block; width: 0;}.MJX-TEX{font-family: MJXZERO, MJXTEX;}.TEX-S1{font-family: MJXZERO, MJXTEX-S1;}@font-face{font-family: MJXZERO; src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Zero.woff") format("woff");}@font-face{font-family: MJXTEX; src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Main-Regular.woff") format("woff");}@font-face{font-family: MJXTEX-S1; src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Size1-Regular.woff") format("woff");}mjx-c.mjx-c46::before{padding: 0.68em 0.653em 0 0; content: "F";}mjx-c.mjx-c28::before{padding: 0.75em 0.389em 0.25em 0; content: "(";}mjx-c.mjx-c78::before{padding: 0.431em 0.528em 0 0; content: "x";}mjx-c.mjx-c29::before{padding: 0.75em 0.389em 0.25em 0; content: ")";}mjx-c.mjx-c3D::before{padding: 0.583em 0.778em 0.082em 0; content: "=";}mjx-c.mjx-c222B.TEX-S1::before{padding: 0.805em 0.61em 0.306em 0; content: "\222B";}mjx-c.mjx-c30::before{padding: 0.666em 0.5em 0.022em 0; content: "0";}mjx-c.mjx-c66::before{padding: 0.705em 0.372em 0 0; content: "f";}mjx-c.mjx-c74::before{padding: 0.615em 0.389em 0.01em 0; content: "t";}mjx-c.mjx-c64::before{padding: 0.694em 0.556em 0.011em 0; content: "d";}mjx-c.mjx-c2032::before{padding: 0.56em 0.275em 0 0; content: "\2032";}mjx-c.mjx-c5B::before{padding: 0.75em 0.278em 0.25em 0; content: "[";}mjx-c.mjx-c61::before{padding: 0.448em 0.5em 0.011em 0; content: "a";}mjx-c.mjx-c2C::before{padding: 0.121em 0.278em 0.194em 0; content: ",";}mjx-c.mjx-c62::before{padding: 0.694em 0.556em 0.011em 0; content: "b";}mjx-c.mjx-c5D::before{padding: 0.75em 0.278em 0.25em 0; content: "]";}mjx-c.mjx-c21D2::before{padding: 0.525em 1em 0.024em 0; content: "\21D2";}define F(x)=∫0xf(t)dtdifferentiate F using fundamental theorem of calculusF′(x)=f(x)since f is continuous on [a,b]⇒F(x) is continuous on [a,b]andF(x) is differentiable on (a,b)Then by lagrange’s mean value theorem ,Explanation:Please refer to solution in this step.Step2/2mjx-container[jax="CHTML"]{line-height: 0;}mjx-container [space="2"]{margin-left: .167em;}mjx-container [space="3"]{margin-left: .222em;}mjx-container [space="4"]{margin-left: .278em;}mjx-container [size="s"]{font-size: 70.7%;}mjx-row{display: table-row;}mjx-row > *{display: table-cell;}mjx-assistive-mml{position: absolute !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0px 0px 0px !important; border: 0px !important; display: block !important; width: auto !important; overflow: hidden !important; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none ... See the full answer