Question

# 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

A
1
B
2
C
12
D
3
Solution
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#### The given equation is:f(x)=2x3−9ax2+12a2x+1To find the extremum points we differentiate and equate it to zero⇒f′(x)=6x2−18ax+12a2f′(x)=0⇒6x2−18ax+12a2=0⇒(x−2a)(x−a)=0⇒x=2a,aNow 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)<0Hence we get a minima at x =2a and and a maxima at x =a. Hence,⇒p=a⇒q=2aUsing the given relation we getp2−q=0⇒a2−2a=0⇒a(a−2)=0⇒a≠0 as a>0⇒a=2 .....Answer

5
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