https://file.aiquickdraw.com/mathbot/file/1726419051824dds1ogl3.jpg

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.

The equation provided is f(x) = ax^2 + bx + 3, with the conditions f'(3) = 4 and f(3) = 45. Find a and b.

Learn how to find the values of a and b in the quadratic equation f(x) = ax^2 + bx + 3, with conditions f'(3) = 4 and f(3) = 45. Explore key algebra and calculus concepts, including derivatives and solving systems of equations. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation