Answer:
x = 2 1/3
Explanation:
You want the x-value of the relative maximum of the cubic f(x) = (x-2)(x-3)².
Observation
The graph of this cubic will touch the x-axis from above at x=3, since that root is the rightmost, and has multiplicity 2. (The graph extends to +∞ to the right of that point.) That is, x=3 is a relative minimum.
The point at x=2 is at the same y-value as the point at x=3. Both have f(x)=0.
The relative extrema of a cubic are symmetrical about its point of inflection. Interestingly, each extreme is at the x-value that is the midpoint between the point of inflection and the point where the graph has a value equal to the other relative extreme.
For this equation, this means the relative maximum is 1/3 of the distance between the roots, at x = 2 +1/3(3 -2) = 2 1/3.
Analytic solution
The derivative of the cubic is ...
f'(x) = (x -3)² +2(x-2)(x -3) = (x -3)(x -3 +2x -4) = (x -3)(3x -7)
The zeros of the derivative are where these factors are zero:
x = 3
x = 7/3
The left-most value corresponds to the relative maximum.
f(x) has a relative maximum at x = 7/3.
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Additional comments
It is clearly easier to "write down" the answer than to go to the trouble of finding the zeros of the derivative. Unfortunately, some explanation is required before one can get to the point where the answer is obvious.
The leading coefficient of the cubic is positive, so its general shape is up to the right. (/) That means the "wiggle" will be a negative slope between two intervals of positive slope. The relative maximum is where the slope changes from positive to negative, at the left side of that "wiggle".
For the derivative, we used the power rule and the product rule:
(u^n)' = n·u^(n-1)·u'
(uv)' = u'v +uv'