Answer :
To solve the problem of finding the function [tex]\( f(x) \)[/tex] given that [tex]\( f'(x) = \frac{8}{\sqrt{x}} \)[/tex] and [tex]\( f(16) = 71 \)[/tex], we follow these steps:
1. Identify the problem: We need to find a function [tex]\( f(x) \)[/tex] whose derivative is [tex]\( \frac{8}{\sqrt{x}} \)[/tex] and then use the given condition [tex]\( f(16) = 71 \)[/tex] to find the particular function.
2. Integrate the derivative: To find [tex]\( f(x) \)[/tex], we need to integrate the given derivative [tex]\( f'(x) = \frac{8}{\sqrt{x}} \)[/tex].
The antiderivative (integral) of [tex]\( \frac{8}{\sqrt{x}} \)[/tex] is [tex]\( 16\sqrt{x} \)[/tex]. When you take the derivative of [tex]\( 16\sqrt{x} \)[/tex], you get back [tex]\( \frac{8}{\sqrt{x}} \)[/tex].
3. Include the constant of integration: When integrating, we introduce a constant of integration [tex]\( C \)[/tex] because the integral of a function is determined up to a constant. Thus, our integrated function looks like:
[tex]\[
f(x) = 16\sqrt{x} + C
\][/tex]
4. Use the initial condition: We are given that [tex]\( f(16) = 71 \)[/tex]. We use this information to find the constant [tex]\( C \)[/tex].
Substitute [tex]\( x = 16 \)[/tex] into the function:
[tex]\[
f(16) = 16\sqrt{16} + C = 71
\][/tex]
Compute [tex]\( 16\sqrt{16} = 16 \times 4 = 64 \)[/tex]. Therefore, the equation becomes:
[tex]\[
64 + C = 71
\][/tex]
Solve for [tex]\( C \)[/tex]:
[tex]\[
C = 71 - 64 = 7
\][/tex]
5. Write the final function: With the constant [tex]\( C \)[/tex] found, the particular solution for [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = 16\sqrt{x} + 7
\][/tex]
This function satisfies both the derivative condition [tex]\( f'(x) = \frac{8}{\sqrt{x}} \)[/tex] and the initial condition [tex]\( f(16) = 71 \)[/tex].
1. Identify the problem: We need to find a function [tex]\( f(x) \)[/tex] whose derivative is [tex]\( \frac{8}{\sqrt{x}} \)[/tex] and then use the given condition [tex]\( f(16) = 71 \)[/tex] to find the particular function.
2. Integrate the derivative: To find [tex]\( f(x) \)[/tex], we need to integrate the given derivative [tex]\( f'(x) = \frac{8}{\sqrt{x}} \)[/tex].
The antiderivative (integral) of [tex]\( \frac{8}{\sqrt{x}} \)[/tex] is [tex]\( 16\sqrt{x} \)[/tex]. When you take the derivative of [tex]\( 16\sqrt{x} \)[/tex], you get back [tex]\( \frac{8}{\sqrt{x}} \)[/tex].
3. Include the constant of integration: When integrating, we introduce a constant of integration [tex]\( C \)[/tex] because the integral of a function is determined up to a constant. Thus, our integrated function looks like:
[tex]\[
f(x) = 16\sqrt{x} + C
\][/tex]
4. Use the initial condition: We are given that [tex]\( f(16) = 71 \)[/tex]. We use this information to find the constant [tex]\( C \)[/tex].
Substitute [tex]\( x = 16 \)[/tex] into the function:
[tex]\[
f(16) = 16\sqrt{16} + C = 71
\][/tex]
Compute [tex]\( 16\sqrt{16} = 16 \times 4 = 64 \)[/tex]. Therefore, the equation becomes:
[tex]\[
64 + C = 71
\][/tex]
Solve for [tex]\( C \)[/tex]:
[tex]\[
C = 71 - 64 = 7
\][/tex]
5. Write the final function: With the constant [tex]\( C \)[/tex] found, the particular solution for [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = 16\sqrt{x} + 7
\][/tex]
This function satisfies both the derivative condition [tex]\( f'(x) = \frac{8}{\sqrt{x}} \)[/tex] and the initial condition [tex]\( f(16) = 71 \)[/tex].
