At what points is the curve y=x/(2x-3)^2 concave up and concave down.? Thank you very much!

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To verify if a function is concave up or concave down, we'll have to do the 2nd derivative test. It means that we have to differentiate the given function twice.

We'll diiferentiate the function:

f'(x) ={(x)'*(2x-3)^2 - (x)*[(2x-3)^2]'}/(2x-3)^4

f'(x) = [(2x-3)^2 - 4x(2x-3)]/(2x-3)^4

f'(x) = (2x-3)(2x - 3 - 4x)/(2x-3)^4

f'(x) = -(2x+3)/(2x-3)^3

Now, we'll differentiate twice, with respect to x:

f''(x) = {[-(2x+3)]'*(2x-3)^3 + -(2x+3)*[(2x-3)^3]'}/(2x-3)^6

f''(x) = {-2*(2x-3)^3 + 6*(2x+3)*[(2x-3)^2]}/(2x-3)^6

f''(x) = (2x-3)^2*[-2*(2x-3) + 6*(2x+3)]/(2x-3)^6

f''(x) = (-4x + 6 + 12x + 18)/(2x-3)^4

f"(x) = (8x + 24)/(2x-3)^4

f"(x) = 8(x+3)/(2x-3)^4

Now, we'll determine the intervals of x values where the function is concave up or concave down.

8(x+3) = 0

x + 3 = 0

x = -3

Since the denominator is always positive, except the value x = 3/2 that cancels it, we'll discuss if f"(x) is positive or negative, considering the x values of numerator.

The 2nd derivative f"(x) < 0, if x is in the range (-infinite,-3) and the function is concave down over the range (-infinite,-3). The 2nd derivative f"(x) > 0, if x is in the range (-3,infinite) and the function is concave up over the range (-3,infinite).

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