Let $f(x) = 4x^3 - 18x^2 + 27x - 5$, $x \in R$. Then which of the following statements are TRUE?

(A) $f''(x) = 24x - 36$

(B) f has local maxima at $x = \frac{3}{2}$ but no minima

(C) f has neither maxima nor minima

(D) f has both maxima and minima

Choose the correct answer from the options given below:

Given $f(x) = 4x^3 - 18x^2 + 27x - 5$, where $x \in R$

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The first derivative:

$f'(x) = 12x^2 - 36x + 27$

The second derivative:

$f''(x) = 24x - 36$

Statement (A) is TRUE.

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To find critical points, set $f'(x) = 0$:

$12x^2 - 36x + 27 = 0$

$4x^2 - 12x + 9 = 0$

$(2x - 3)^2 = 0$

$x = \frac{3}{2}$

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Testing $x = \frac{3}{2}$ using the second derivative:

$f''\left(\frac{3}{2}\right) = 24\left(\frac{3}{2}\right) - 36$

$f''\left(\frac{3}{2}\right) = 36 - 36$

$f''\left(\frac{3}{2}\right) = 0$

The second derivative test is inconclusive.

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Analyzing the sign of $f'(x)$:

$f'(x) = 12\left(x - \frac{3}{2}\right)^2$

For $x < \frac{3}{2}$: $f'(x) = 12 \times (\text{positive})^2 > 0$, function is increasing

At $x = \frac{3}{2}$: $f'(x) = 0$

For $x > \frac{3}{2}$: $f'(x) = 12 \times (\text{positive})^2 > 0$, function is increasing

Since the function is increasing on both sides of $x = \frac{3}{2}$, this point is a horizontal point of inflection, not a maximum or minimum.

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Statement (A): $f''(x) = 24x - 36$ is TRUE

Statement (B): f has local maxima at $x = \frac{3}{2}$ is FALSE

Statement (C): f has neither maxima nor minima is TRUE

Statement (D): f has both maxima and minima is FALSE

Therefore, statements (A) and (C) are TRUE.