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# It is given that for the function $f(x)=x^3-6x^2+px+q\;on\;[1,3].$Rolles theorem holds with $c=2+\large\frac{1}{\sqrt 3}.$Find the values of p and q.

Toolbox:
• As per Rolle's theorem
• $f(a)=f(b) \: and \: f'(c) = 0$
$f(1) = -5+p+q$
$f(3) = -27+3p+q$
$f'(x) = 3x^2-12x+p$
$f' \bigg(2+\large\frac{1}{\sqrt 3} \bigg) = p-1=0$
p=1
$q-4=q-24=0$
$\Rightarrow$ q is any real number.

edited Mar 25, 2013