answer:To find the points of inflection of a function, you need to find the second derivative of the function and determine where it changes sign. The given function is: [y = frac{4x^2 - 3}{x^3}] First, let's find the first derivative (frac{dy}{dx}). For that, apply the quotient rule which is, if (f(x) = frac{g(x)}{h(x)}), then (f'(x) = frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}). Here, (g(x) = 4x^2 - 3) and (h(x) = x^3), so [g'(x) = frac{d}{dx}(4x^2 - 3) = 8x] and [h'(x) = frac{d}{dx}(x^3) = 3x^2.] Thus, [frac{dy}{dx} = frac{(8x)(x^3) - (4x^2 - 3)(3x^2)}{x^6} = frac{8x^4 - 12x^4 + 9}{x^6} = frac{-4x^4 + 9}{x^6}.] Now, let's simplify and find the second derivative: [frac{dy}{dx} = -frac{4x^4 - 9}{x^6} = -4x^{-2} + 9x^{-6}.] The second derivative (frac{d^2y}{dx^2}) involves differentiating (-4x^{-2} + 9x^{-6}) with respect to (x). Use the power rule, which says if (f(x) = x^n), then (f'(x) = nx^{n-1}). [frac{d^2y}{dx^2} = frac{d}{dx}(-4x^{-2} + 9x^{-6}) = 8x^{-3} - 54x^{-7}.] Now we set the second derivative equal to zero to find potential inflection points: [8x^{-3} - 54x^{-7} = 0.] Multiplying every term by (x^7) to clear the negative powers: [8x^4 - 54 = 0.] [8x^4 = 54.] [x^4 = frac{54}{8} = frac{27}{4}.] [x = left(frac{27}{4}right)^{frac{1}{4}}.] [x = sqrt[4]{frac{27}{4}} = sqrt[4]{27} / sqrt[4]{4} = frac{3}{sqrt[2]{2}} = frac{3}{2^{frac{1}{2}}} = frac{3sqrt{2}}{2}.] Hence, we have (x = frac{3sqrt{2}}{2}) as the place where there might be an inflection point. To confirm if this is truly an inflection point, we would check the sign of the second derivative just before and just after this value, if it changes sign then (x = frac{3sqrt{2}}{2}) is indeed an inflection point. Given the steps, the calculations to check the sign change are extensive but follow naturally by replacing (x) in the simplified second derivative and observing its behavior around (x = frac{3sqrt{2}}{2}).

answer:Let's correct the mistake and re-calculate the first derivative of the function [y = frac{4x^2 - 3}{x^3}]. Starting again with the quotient rule, which is given as [f'(x) = frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}] for a function [f(x) = frac{g(x)}{h(x)}], where, in our case, [g(x) = 4x^2 - 3] and [h(x) = x^3]. First, calculate the derivatives of (g(x)) and (h(x)): [g'(x) = frac{d}{dx}(4x^2 - 3) = 8x,] [h'(x) = frac{d}{dx}(x^3) = 3x^2.] Using the quotient rule: [f'(x) = frac{(8x)x^3 - (4x^2 - 3)(3x^2)}{x^6}] [= frac{8x^4 - (12x^4 - 9)}{x^6}] [= frac{8x^4 - 12x^4 + 9}{x^6}] [= frac{-4x^4 + 9}{x^6}] Simplifying, we get: [f'(x) = -frac{4x^4}{x^6} + frac{9}{x^6}] [= -4x^{-2} + 9x^{-6}] It seems there was a misinterpretation in my steps, but this clarification aligns with the correction process. The last expression is correctly derived from the quotient rule but was not needed for the next step of finding the second derivative. To correct further steps, let's directly calculate the first derivative again correctly for clarity: [ y = frac{4x^2 - 3}{x^3} ] [ y' = frac{(8x)x^3 - (4x^2 - 3)3x^2}{(x^3)^2} ] [ y' = frac{8x^4 - 12x^4 + 9}{x^6} ] [ y' = frac{-4x^4 + 9}{x^6} ] Now, simplify and differentiate again: [ y' = -4x^{-2} + 9x^{-6} ] Noticing that my interpretation remained consistent, the clarification didn't alter the derivative calculation but did seek to correct the approach. The next step, indeed, would be to calculate the second derivative directly from the correctly stated first derivative, [y' = -4x^{-2} + 9x^{-6}], followed by a reassessment of any errors in detailing the solution. However, let's correct the approach towards finding the second derivative based on the final expression of the first derivative, as the misunderstanding seemed to have stemmed from the explanation post-first derivative. Given the confusion, the proper approach was to identify and correct the mishandled components, which involved re-confirming the steps accurately for the second derivative calculation. My earlier response miscalculated this due to a breakdown in applying the conceptual steps thoroughly. Let's properly approach the calculation for the second derivative, reaffirming the necessity for accuracy in mathematical derivations.

answer:To find the derivative of the function given in the form ((4x^2 - 3)x^{-3}), we'll use the product rule since the function is in the form of (f(x)g(x)), where (f(x) = 4x^2 - 3) and (g(x) = x^{-3}). The product rule is stated as ((fg)' = f'g + fg'). First, calculate the derivatives of (f(x)) and (g(x)): - (f'(x) = frac{d}{dx}(4x^2 - 3) = 8x), - (g'(x) = frac{d}{dx}(x^{-3}) = -3x^{-4}). Using the Product rule: - ((fg)' = (4x^2 - 3)'x^{-3} + (4x^2 - 3)(x^{-3})'), - ((fg)' = (8x)x^{-3} + (4x^2 - 3)(-3x^{-4})), - Simplify to get: [ (fg)' = 8x^{-2} - 12x^{-4} + 9x^{-4} ] [ (fg)' = 8x^{-2} - 3x^{-4} ] So, the derivative of the function (f(x)g(x) = (4x^2 - 3)x^{-3}) with respect to (x) is (8x^{-2} - 3x^{-4}).

answer:Let's correctly find the derivative for the function given as [f(x) = (4x^2 - 3)x^{-3}.] Given my oversight, I appreciate your patience. Let's simplify the function first before differentiating, which can sometimes make the differentiation process more straightforward: [f(x) = 4x^2x^{-3} - 3x^{-3} = 4x^{-1} - 3x^{-3}.] Now let's differentiate (f(x)) with respect to (x): [f'(x) = frac{d}{dx}[4x^{-1}] - frac{d}{dx}[3x^{-3}].] Using the power rule for differentiation, which states that ( frac{d}{dx}[x^n] = nx^{n-1}), we find: [f'(x) = 4(-1)x^{-1-1} - 3(-3)x^{-3-1}] [= -4x^{-2} + 9x^{-4}.] Therefore, the correct derivative of (f(x) = (4x^2 - 3)x^{-3}) is: [f'(x) = -4x^{-2} + 9x^{-4}.] Apologies for the confusion earlier, and thank you for your patience.