Suppose x_1 and x_2 are the point of maximum | vedclass

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(B) Given $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$.Find the derivative: $f'(x) = 6x^2 - 18ax + 12a^2 = 6(x^2 - 3ax + 2a^2) = 6(x - a)(x - 2a)$.Setting $f'(x

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$2$

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(B) Given $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$.Find the derivative: $f'(x) = 6x^2 - 18ax + 12a^2 = 6(x^2 - 3ax + 2a^2) = 6(x - a)(x - 2a)$.Setting $f'(x) = 0$ gives critical points $x = a$ and $x = 2a$.Find the second derivative: $f''(x) = 12x - 18a$.Check the nature of critical points:$f''(a) = 12a - 18a = -6a$$f''(2a) = 12(2a) - 18a = 6a$Case $1$: If $a > 0$,then $f''(a)  0$ (minimum at $x_2 = 2a$).Given $x_1^2 = x_2$,we have $a^2 = 2a$. Since $a > 0$,$a = 2$.Case $2$: If $a  0$ (minimum at $x_2 = a$) and $f''(2a) Given $x_1^2 = x_2$,we have $(2a)^2 = a$,which means $4a^2 = a$. Since $a Thus,the value of $a$ is $2$.

Suppose $x_1$ and $x_2$ are the point of maximum and the point of minimum respectively of the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$. If the equality $x_1^2 = x_2$ holds true,then the value of $a$ must be:

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