Let's solve the initial value problem step by step.
1. Determine the indefinite integral of [tex]\( f'(x) \)[/tex]:
We start with the derivative of the function, [tex]\( f'(x) = 6x - 5 \)[/tex]. To find [tex]\( f(x) \)[/tex], we need to integrate [tex]\( f'(x) \)[/tex] with respect to [tex]\( x \)[/tex].
[tex]\[
f(x) = \int (6x - 5) \, dx
\][/tex]
2. Compute the integral:
To integrate [tex]\( 6x - 5 \)[/tex]:
[tex]\[
\int (6x - 5) \, dx = \int 6x \, dx - \int 5 \, dx
\][/tex]
Now handle each term separately:
[tex]\[
\int 6x \, dx = 6 \int x \, dx = 6 \left( \frac{x^2}{2} \right) = 3x^2
\][/tex]
[tex]\[
\int 5 \, dx = 5x
\][/tex]
Putting it all together:
[tex]\[
f(x) = 3x^2 - 5x + C
\][/tex]
Here, [tex]\( C \)[/tex] is the constant of integration.
3. Use the initial condition to determine [tex]\( C \)[/tex]:
We are given the initial condition [tex]\( f(0) = 9 \)[/tex].
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( f(0) = 9 \)[/tex] into the equation [tex]\( f(x) = 3x^2 - 5x + C \)[/tex]:
[tex]\[
f(0) = 3(0)^2 - 5(0) + C = 9
\][/tex]
Simplifying, we have:
[tex]\[
C = 9
\][/tex]
4. Write the final solution:
Substitute [tex]\( C \)[/tex] back into the function:
[tex]\[
f(x) = 3x^2 - 5x + 9
\][/tex]
Therefore, the solution to the initial value problem is:
[tex]\[
f(x) = \boxed{3x^2 - 5x + 9}
\][/tex]
