Q.Verify Lagrange's mean value theorem for the function $f(x) = (x - 1)(x - 2)(x - 3)$ on $[0, 4]$.

Expanding: $f(x)=(x-1)(x-2)(x-3) = x^3-6x^2+11x-6$, a polynomial, so continuous and differentiable everywhere — LMVT applies on $[0,4]$.

$f(0)=-6$. $f(4)=64-96+44-6=6$. Chord slope $=\dfrac{6-(-6)}{4-0}=\dfrac{12}{4}=3$.

$f'(x)=3x^2-12x+11$. Set $f'(c)=3$: $3c^2-12c+11=3 \Rightarrow 3c^2-12c+8=0 \Rightarrow c=\dfrac{12\pm\sqrt{144-96}}{6}=\dfrac{12\pm\sqrt{48}}{6}=2\pm\dfrac{2\sqrt3}{3}$. …

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