Question
If the function f(x)=2x3−9ax2+12a2 x+1, where a > 0, attains its maximum and minimum at p and q respectively such that p2=q, then a equals
Answer: Option C
:
C
We have, f(x)=2x3−9ax2+12a2x+1∴f(x)=6x2−18ax+12a2=0⇒6[x2−3ax+2a2]=0⇒x2−3ax+2a2=0⇒x2−2ax−ax+2a2=0⇒x(x−2a)−a(x−2a)=0⇒(x−a)(x−2a)=0⇒x=a,x=2a
Now, f′(x)=12x−18a
∴f′(a)=12a−18a=−6a<0∴f(x) will be maximum at x = a
i.e. p = a
Also, f′(2a)=24a−18a=6a∴f(x)will be minimum at x = 2a
i.e.q = 2a
Given, p2=q
⇒a2=2a⇒a=2.
Hence (c) is the correct answer.
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:
C
We have, f(x)=2x3−9ax2+12a2x+1∴f(x)=6x2−18ax+12a2=0⇒6[x2−3ax+2a2]=0⇒x2−3ax+2a2=0⇒x2−2ax−ax+2a2=0⇒x(x−2a)−a(x−2a)=0⇒(x−a)(x−2a)=0⇒x=a,x=2a
Now, f′(x)=12x−18a
∴f′(a)=12a−18a=−6a<0∴f(x) will be maximum at x = a
i.e. p = a
Also, f′(2a)=24a−18a=6a∴f(x)will be minimum at x = 2a
i.e.q = 2a
Given, p2=q
⇒a2=2a⇒a=2.
Hence (c) is the correct answer.
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