$\text { Find the cubic function of the form } f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d \text {, where } a \neq 0 \text { and the }$ coefficients $a , b , c , d$ are real numbers, which satisfies the conditions given below.
Relative maximum: $( 3,0 )$
Relative minimum: $( 5 , - 2 )$
Inflection point: $( 4 , - 1 )$
A) $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 6 x ^ { 2 } + \frac { 45 } { 2 } x - 27$
B) $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 6 x ^ { 2 } + \frac { 41 } { 2 } x - 26$
C) $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 6 x ^ { 2 } + \frac { 45 } { 2 } x + 27$
D) $f ( x ) = - \frac { 1 } { 2 } x ^ { 3 } + 6 x ^ { 2 } - \frac { 45 } { 2 } x + 27$
E) $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 6 x ^ { 2 } + \frac { 41 } { 2 } x + 26$

Question

Quizplus · Accepted Answer

To find the correct cubic function, we need to use the given points to determine the coefficients. The first derivative of a cubic function $f(x) = ax^3 + bx^2 + cx + d$ is $f'(x) = 3ax^2 + 2bx + c$, which gives us the slope at any point x. The second derivative, $f''(x) = 6ax + 2b$, tells us about the concavity of the function.1. **Relative Maximum at (3,0)** and **Relative Minimum at (5,-2)** imply that $f'(3) = 0$ and $f'(5) = 0$ because the slope of the tangent at these points is zero.2. **Inflection Point at (4,-1)** implies that $f''(4) = 0$ because the concavity of the function changes at this point.Given these conditions, we can set up equations based on the derivatives, but without doing the detailed algebra, we can check the given options directly by plugging in the points to see which function satisfies all the conditions:- For the relative maximum and minimum, the function must pass through the points (3,0) and (5,-2).- For the inflection point, the second derivative of the function must change sign at $x = 4$, but more directly, the function must pass through (4,-1).Checking the options, **Option A** satisfies all these conditions:- It passes through (3,0) and (5,-2), satisfying the conditions for the relative maximum and minimum.- It also passes through (4,-1), consistent with the inflection point condition.Thus, without needing to solve the system of equations explicitly, we can determine that **Option A** is the correct cubic function that meets all the given conditions.

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$\text { Find the cubic function of the form } f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d \text {, where } a \neq 0 \text { and the }$