To find the solution to the initial value problem, we need to follow these steps:
1. Integrate the given derivative [tex]\( f'(x) \)[/tex] to find the general form of the function [tex]\( f(x) \)[/tex].
2. Determine the constant of integration using the given initial condition.
Let's start by integrating the derivative:
Given:
[tex]\[ f'(x) = 8x - 5 \][/tex]
Step 1: Integrate the given derivative
To find [tex]\( f(x) \)[/tex], we need to integrate [tex]\( f'(x) \)[/tex]:
[tex]\[ f(x) = \int (8x - 5) \, dx \][/tex]
Performing the integration gives us:
[tex]\[ f(x) = \int 8x \, dx - \int 5 \, dx \][/tex]
Now calculate each integral separately:
[tex]\[ \int 8x \, dx = 8 \cdot \frac{x^2}{2} = 4x^2 \][/tex]
[tex]\[ \int 5 \, dx = 5x \][/tex]
Putting these results together:
[tex]\[ f(x) = 4x^2 - 5x + C \][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
Step 2: Determine the constant [tex]\( C \)[/tex] using the initial condition
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]\[ 9 = 4(0)^2 - 5(0) + C \][/tex]
[tex]\[ 9 = C \][/tex]
So, the constant [tex]\( C \)[/tex] is 9.
Final Solution:
Substituting [tex]\( C \)[/tex] back into the general solution gives us the specific solution:
[tex]\[ f(x) = 4x^2 - 5x + 9 \][/tex]
Hence, the solution to the initial value problem is:
[tex]\[ f(x) = 4x^2 - 5x + 9 \][/tex]
