Question

Verify Mean Value Theorem,if \(f(x)=x^2-4x-3\) in the interval \([a,b]\),where \(a=1\) and \(b=4\)

Answer 1

The given function is \(f(x)=x^2-4x-3\)
\(f\), being a polynomial function, is continuous in \([1,4]\) and is differentiable in \((1,4)\) whose derivative is \(2x−4\).
\(f(1)=1^2-4\times1-3=-6\)
\(f(4)=(4)^2-4\times4-3=-3\)
\(∴\frac{f(b)-f(a)}{b-a}=\frac{f(4)-f(1)}{4-1}\)
\(=\frac{-3-(-6)}{3}=\frac{3}{3}=1\)
Mean Value Theorem states that there is a point \(c∈(1,4)\) such that \(f'(c)=1\)
\(f'(c)=1\)
\(⇒2c-4=1\)
\(⇒c=\frac{5}{2}\),where \(c=\frac{5}{2}∈(1,4)\)
Hence, Mean Value Theorem is verified for the given function.