Given that the cubic function is of the form:

$f(x) = ax^3 + bx^2 + cx + d$

The stationary point occurs where the derivative of the function is zero. Let's break down the information step by step.

Step 1: Use the condition of the stationary point

Since the function has a stationary point at $(2, 5)$, the derivative at $x = 2$ must be zero:

$f'(x) = 3ax^2 + 2bx + c$

At $x = 2$, we know:

$f'(2) = 0 \implies 3a(2)^2 + 2b(2) + c = 0$ Simplifying this equation:

$3a(4) + 2b(2) + c = 0 \implies 12a + 4b + c = 0 \quad \text{(Equation 1)}$

Additionally, since $f(2) = 5$, we substitute $x = 2$ into the cubic function itself:

$f(2) = a(2)^3 + b(2)^2 + c(2) + d = 5$ Simplifying:

$a(8) + b(4) + c(2) + d = 5 \implies 8a + 4b + 2c + d = 5 \quad \text{(Equation 2)}$

Step 2: Use the point $(1, 3)$

The cubic function also passes through $(1, 3)$, so:

$f(1) = a(1)^3 + b(1)^2 + c(1) + d = 3$ Simplifying:

$a + b + c + d = 3 \quad \text{(Equation 3)}$

Step 3: Solve the system of equations

Now, we have the following system of equations:

1. $12a + 4b + c = 0$
2. $8a + 4b + 2c + d = 5$
3. $a + b + c + d = 3$

We can now solve these equations to find $a$, $b$, $c$, and $d$.

Solving Equation 1 for $c$:

$c = -12a - 4b$

Substituting $c = -12a - 4b$ into Equation 2:

$8a + 4b + 2(-12a - 4b) + d = 5$ Simplifying:

$8a + 4b - 24a - 8b + d = 5 \implies -16a - 4b + d = 5$ So, we have:

$d = 16a + 4b + 5 \quad \text{(Equation 4)}$

Substituting $c = -12a - 4b$ and $d = 16a + 4b + 5$ into Equation 3:

$a + b + (-12a - 4b) + (16a + 4b + 5) = 3$ Simplifying:

$a + b - 12a - 4b + 16a + 4b + 5 = 3 \implies 5a + b + 5 = 3$ So:

$5a + b = -2 \quad \text{(Equation 5)}$

Step 4: Solve for $a$ and $b$

From Equation 5:

$b = -2 - 5a$

Substitute this into Equation 1:

$12a + 4(-2 - 5a) + c = 0 \implies 12a - 8 - 20a + c = 0 \implies -8a - 8 + c = 0$ So:

$c = 8a + 8$

Now, we substitute $b = -2 - 5a$ and $c = 8a + 8$ into Equation 4:

$d = 16a + 4(-2 - 5a) + 5 \implies d = 16a - 8 - 20a + 5 \implies d = -4a - 3$

Final Values:

Thus, the values of the constants are:

• $a = -\frac{2}{5}$
• $b = -\frac{14}{5}$
• $c = \frac{8}{5}$
• $d = -\frac{11}{5}$

Do you have any questions or want more details?

Related Questions:

1. How do you find stationary points of higher-order polynomials?
2. What is the significance of the second derivative at a stationary point?
3. Can a cubic function have more than one stationary point?
4. How can you determine if a stationary point is a local maximum or minimum?
5. What is the role of the constant $d$ in a cubic function?

Tip:

Always check the nature of stationary points using the second derivative to confirm if it's a maximum, minimum, or inflection point.