Final answer:
The question involves finding coefficients of a cubic function with both a relative minimum and inflection point at x=2, using the first and second derivative tests and possibly the quadratic formula and integral calculations for curve analysis.
Step-by-step explanation:
The subject of the question pertains to the analysis of a cubic function, specifically its relative minimum and inflection point. Considering the function f(x)=x³+bx²+cx+d, it is stated that this function has both a relative minimum and an inflection point at x=2. This scenario implies that both the first and second derivatives of the function will have specific characteristics at x=2. Specifically, the first derivative, f'(x), would be zero at x=2 for a relative minimum or maximum, and the second derivative, f''(x), would be zero at x=2 for an inflection point.
Since the function has both of these features at the same point, one can set up a system of equations using f'(x) and f''(x) to derive values for the coefficients b, c, and d in the cubic function that satisfy these conditions. We can then employ techniques like quadratic formula to solve related optimization and curve analysis problems, as well as understanding the integral of the function to find the area under the curve between two points. The second derivative test may also be applied to determine the nature of the critical point at x=2.