The equation in the image provided is:

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

The conditions given are:

1. $f'(3) = 4$
2. $f(3) = 45$

We need to find the values of $a$ and $b$.

Step 1: Derive the equation for $f'(x)$

The derivative of $f(x)$ with respect to $x$ is:

$f'(x) = 2ax + b$

Step 2: Use the given condition $f'(3) = 4$

Substitute $x = 3$ into $f'(x)$:

$f'(3) = 2a(3) + b = 4$ $6a + b = 4 \quad \text{(Equation 1)}$

Step 3: Use the given condition $f(3) = 45$

Substitute $x = 3$ into $f(x)$:

$f(3) = a(3^2) + b(3) + 3 = 45$ $9a + 3b + 3 = 45$ $9a + 3b = 42 \quad \text{(after subtracting 3 from both sides)}$ $3a + b = 14 \quad \text{(Equation 2, after dividing by 3)}$

Step 4: Solve the system of equations

We have two equations:

1. $6a + b = 4$
2. $3a + b = 14$

Now, subtract Equation 2 from Equation 1:

$(6a + b) - (3a + b) = 4 - 14$ $3a = -10$ $a = -\frac{10}{3}$

Step 5: Substitute $a$ back into one of the equations

Substitute $a = -\frac{10}{3}$ into Equation 2:

$3\left(-\frac{10}{3}\right) + b = 14$ $-10 + b = 14$ $b = 24$

Final Answer:

• $a = -\frac{10}{3}$
• $b = 24$

Would you like more details or have any questions? Here are five related questions to explore:

1. How would you graph the function $f(x) = ax^2 + bx + 3$ with the values of $a$ and $b$ found?
2. What does the derivative $f'(x) = 2ax + b$ represent in terms of the slope of the curve?
3. How would the value of $f'(x)$ change if $a$ and $b$ were different?
4. Can you determine the vertex of the quadratic function using the values of $a$ and $b$?
5. How would you solve for $a$ and $b$ if the function was cubic instead of quadratic?

Tip:

When solving systems of equations, substitution or elimination methods can both be effective.