Answer:
2a. maximum: (1, 8); minimum: (3, 0)
2b. point of inflection: (2, 4)
3. a = 1/4
Step-by-step explanation:
You want the extrema and the point of inflection of f(x) = 2x³ -12x² +18x, and you want the value of 'a' that makes the end behavior a horizontal asymptote at y = 1.
2. Cubic
The cubic function f(x) = 2x³ -12x² +18x has derivative ...
f'(x) = 6x² -24x +18 = 6(x² -4x +3) = 6(x -1)(x -3)
The derivative will be zero at x = 1 and 3. These are the x-coordinates of the extrema of f(x).
The values of f(x) at those points are ...
f(x) = 2x((x -6)x +9)
f(1) = 2·1((1 -6)1 +9) = 2(-5 +9) = 8
f(3) = 2·3(3 -6)·3 +9 = 6(-9 +9) = 0
The leading coefficient is positive, so the leftmost portion of the curve is increasing. The maximum will be at the leftmost turning point. The minimum is the other one (rightmost).
The maximum is (1, 8); the minimum is (3, 0).
The point of inflection is the midpoint between the extrema of a cubic:
((1, 8) +(3, 0))/2 = (2, 4) . . . . point of inflection
3. Limit
You want the limit as x → -∞ of ((x -2)(3x² +5) -4ax³)/(8ax³ +5x -1) to be 1.
The ratio of leading terms of the numerator and denominator is ...
(3 -4a)x³/(8ax³) = 1
This requires ...
3 -4a = 8a ⇒ 3 = 12a ⇒ a = 1/4
The value of 'a' that makes the limit be 1 is 1/4.
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Additional comments
A cubic is symmetrical about its point of inflection, so the point of inflection will be equidistant from the extrema, their midpoint. When the leading coefficient is positive, the function is increasing everywhere except between the local extrema. (The first attachment has a graph of the derivative, for reference.)
The end behavior of a rational function of equal numerator and denominator degree is the ratio of the leading coefficients.
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