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If the function f(x)=2x3−9ax2+12a2x+1, where a>0, attains its maximum and minimum at p and q respectively such that p2=q then a equals to

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The given equation is:

f(x)=2x3−9ax2+12a2x+1

To find the extremum points we differentiate and equate it to zero

⇒f′(x)=6x2−18ax+12a2

f′(x)=0

⇒6x2−18ax+12a2=0

⇒(x−2a)(x−a)=0

⇒x=2a,a

Now to find whether at the critical points we find a maxima or minima we use the second derivative test.

⇒f′′(x)=12x−18a

⇒f′′(2a)=24a−18a=6a

⇒f′′(2a)>0

⇒f′′(a)=12a−18a=−6a

⇒f′′(a)<0

Hence we get a minima at x =2a and and a maxima at x =a. Hence,

⇒p=a

⇒q=2a

Using the given relation we get

p2−q=0

⇒a2−2a=0

⇒a(a−2)=0

⇒a≠0 as a>0

⇒a=2 .....Answer

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