QJEEM03Maa2c2: JEE Main 2003 Mathematics - Class 12 JEEM03Maa2c2 Solution: If the function where attains its maximum and minimum at and…

Question

Verified Solution • 2025–26 Session moderate

If the function

f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,

where

a&gt;0,

attains its maximum and minimum at

p

and

q

respectively such that

{p^2} = q

, then

a

equals

Accepted Answer

f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1

f'\left( x \right) = 6{x^2} - 18ax + 12{a^2};

\,\,\,\,\,\,\,\,\,\,f''\left( x \right) = 12x - 18a

For max. or min.

6{x^2} - 18ax + 12{a^2} = 0

\Rightarrow {x^2} - 3ax + 2{a^2} = 0

\Rightarrow x = a

or

x=2a.

At

x=a

f''(a) = 12a - 18a = -6a &lt; 0

At

x=a

maximum As f''(

a

) < 0.

At

x=2a

f''(a) = 24a - 18a = 6a &gt; 0

At

x=2a

minimum As f''(2

a

) > 0.

\therefore

p=a

and

q=2a

As per question

{p^2} = q

\therefore

{a^2} = 2a

\Rightarrow

a(a - 2) = 0

\therefore

a

= 0, 2

but

a &gt; 0,

therefore,

a=2.

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