How to do this question?---Let
$g : R \to R, g(x) = ax^3 + bx^2 + c$ where $a, b, c \in R$
and
$h : R \to R, h(x) = \frac{x^2}{4}$
The graph of $y = g(x)$ passes through the point $(0, 9)$.
The graphs of $y = g(x)$ and $y = h(x)$ pass through the point $(6, 9)$ and have the same gradient at this point.
The graphs of $y = g(x)$ and $y = h(x)$ are shown below.
**Chart/Diagram Description:**
* **Type:** Coordinate plane with two functions plotted.
* **Coordinate Axes:**
* X-axis labeled 'x' points horizontally to the right. The origin O is marked at (0, 0).
* Y-axis labeled 'y' points vertically upwards.
* **Functions:**
* A dashed curve labeled 'y = g(x)' starts on the left, goes through approximately (0, 9), then curves downwards before possibly curving upwards further to the right. It passes through the point (6, 9).
* A solid curve labeled 'y = h(x)' starts at the origin (0, 0), curves upwards, representing a parabola opening upwards. It passes through the point (6, 9).
* **Points:**
* The point (0, 9) is marked on the Y-axis and lies on the dashed curve y = g(x).
* The point (6, 9) is marked with a solid circle and lies on both the dashed curve y = g(x) and the solid curve y = h(x). A dashed line segment extends from (6, 9) upwards, tangent to both curves at that point, indicating they have the same gradient.
* The origin O is marked at (0, 0), and the solid curve y = h(x) appears to pass through it.
Find the values of $a$, $b$ and $c$.
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We have a function parameter problem. Given two functions: g of x equals a x cubed plus b x squared plus c, and h of x equals x squared divided by four. We know that g of x passes through point zero comma nine, both functions pass through point six comma nine, and they have the same gradient at that point. We need to find the values of parameters a, b, and c.
Let's start by using the first condition. Since g of x passes through point zero comma nine, we substitute x equals zero into the function. g of zero equals a times zero cubed plus b times zero squared plus c, which simplifies to c equals nine. So we have found our first parameter: c equals nine.
Now we use the conditions at point six comma nine. Both functions pass through this point and have the same gradient. First, let's find the derivatives. g prime of x equals three a x squared plus two b x, and h prime of x equals x divided by two. At x equals six, h prime of six equals three. Therefore, g prime of six equals one hundred eight a plus twelve b, which must equal three.
Now we solve the system of equations. From g of six equals nine, we get two hundred sixteen a plus thirty six b equals zero, which simplifies to six a plus b equals zero. From the gradient condition, we have nine a plus b equals one fourth. Subtracting the first equation from the second gives us three a equals one fourth, so a equals one twelfth. Substituting back, b equals negative one half. Therefore, our final answer is a equals one twelfth, b equals negative one half, and c equals nine.
让我们从第一个条件开始。由于g函数经过点零逗号九，我们将x等于零代入函数。g零等于a乘以零的三次方加b乘以零的二次方加c，这简化为c等于九。所以我们找到了第一个参数：c等于九。
现在我们使用点六逗号九处的条件。两个函数都经过这个点并且有相同的梯度。首先，让我们求导数。g的导数等于三a x的平方加二b x，h的导数等于x除以二。在x等于六时，h的导数等于三。因此，g在六处的导数等于一百零八a加十二b，这必须等于三。
现在我们解这个方程组。从g六等于九，我们得到二百一十六a加三十六b等于零，简化为六a加b等于零。从梯度条件，我们有九a加b等于四分之一。第二个方程减去第一个方程得到三a等于四分之一，所以a等于十二分之一。代入回去，b等于负二分之一。因此，我们的最终答案是a等于十二分之一，b等于负二分之一，c等于九。
总结我们学到的内容：我们通过利用已知点条件确定函数参数，通过导数相等条件建立方程组，然后系统求解线性方程组得到最终答案。最终结果是a等于十二分之一，b等于负二分之一，c等于九。