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MA441- LEC 7 -1.8 The Intermediate Value Theorem (IVT)

By KWANG

Goal

  • The Intermediate Value Theorem (IVT)

The Intermediate Value Theorem (IVT)

Suppose that \( `f\) is continus on the closed interval \([a,b]\) and let \(N\) be any number between \(f(a)\) and \(f(b)\). where \(f(a)\neq f(b)\) . Then there exists a number \(c\) in \((a,b)\) such that \(f(c)=N\).

Remark

  1. \(c\) does not need to be unique.
  2. Between \(a\) and \(b\) means \((a,b)\) if \(a<b\).
  3. close interval \([a,b]\) in \(\mathbb{R}\) is very important. Real numbers have very good properties like compact and connected(path-connected) and continuous function \(f\) preserves it. So \(f([a,b])\) is compact and connected in realine and it is also close interval. If we assume \(f(a)<f(b)\), \(N\in (f(a),f(b))\subset f([a,b])\).
  4. IVT is very useful to prove the existence of a root of certain equations.

Example 1.

Show that there is a root of the equation \[ 4x^3-6x^2+3x-2=0 \] between \(1\) and \(2\).

How to.

  1. Let \(f(x)=4x^3-6x^2+3x-2\) then \(f\) is continuous on \(\mathbb{R}\)
  2. A root \(c\) of \(f(x)=0\) means \(f(c)=0\). Therefore \(N=0\).
  3. Since \(c\in (1,2)\), we need to choose \(a=1\), \(b=2\).
  4. We only need to check whether \(N=0\) is between \(f(a)\) and \(f(b)\).


Eventually, we need to prove

a. \(f(a)>0\) and \(~f(b)<~0\) or \(~f(a)<~0\) and \(~f(b)>0\)

Check \(f(a)\) and \(f(b)\).

Example 2. Use IVT to show that there is a root of the given equation in the specified interval.

  1. \(x^4+x-3=0\), \((1,2)\)
  2. \(\cos{x}=x\), \((0,1)\)