If the function f(x) = 2x^3 - 9ax^2 + 12a^2x | vedclass

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(C) Given $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$.Differentiating $f(x)$ with respect to $x$,we get $f'(x) = 6x^2 - 18ax + 12a^2$.For maximum or minimum,we

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$2$

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(C) Given $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$.Differentiating $f(x)$ with respect to $x$,we get $f'(x) = 6x^2 - 18ax + 12a^2$.For maximum or minimum,we set $f'(x) = 0$:$6(x^2 - 3ax + 2a^2) = 0$$6(x - a)(x - 2a) = 0$Thus,the critical points are $x = a$ and $x = 2a$.Now,we find the second derivative $f''(x) = 12x - 18a$.At $x = a$,$f''(a) = 12a - 18a = -6a$. Since $a > 0$,$f''(a) At $x = 2a$,$f''(2a) = 12(2a) - 18a = 6a$. Since $a > 0$,$f''(2a) > 0$,so $f(x)$ has a minimum at $q = 2a$.Given the condition $p^2 = q$,we substitute the values:$a^2 = 2a$$a^2 - 2a = 0$$a(a - 2) = 0$Since $a > 0$,we must have $a = 2$.

If the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$,where $a > 0$,attains its maximum and minimum at $p$ and $q$ respectively such that $p^2 = q$,then $a$ equals

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