Answer :

To solve the problem and find the function [tex]\( f(x) \)[/tex] given that its derivative is [tex]\( f^{\prime}(x) = \frac{8}{\sqrt{x}} \)[/tex] and [tex]\( f(16) = 71 \)[/tex], follow these steps:

1. Integrate the derivative to find [tex]\( f(x) \)[/tex]:

The given derivative is [tex]\( f^{\prime}(x) = \frac{8}{\sqrt{x}} \)[/tex]. To find the original function [tex]\( f(x) \)[/tex], we need to integrate [tex]\( f^{\prime}(x) \)[/tex].

[tex]\[
f(x) = \int \frac{8}{\sqrt{x}} \, dx
\][/tex]

Rewrite the integrand: [tex]\(\frac{8}{\sqrt{x}} = 8x^{-\frac{1}{2}}\)[/tex].

Now, integrate:
[tex]\[
f(x) = \int 8x^{-\frac{1}{2}} \, dx = 8 \cdot 2x^{\frac{1}{2}} + C = 16x^{\frac{1}{2}} + C
\][/tex]

Here, [tex]\( C \)[/tex] is the constant of integration.

2. Use the initial condition to find [tex]\( C \)[/tex]:

We're given that [tex]\( f(16) = 71 \)[/tex]. Substitute [tex]\( x = 16 \)[/tex] into the equation for [tex]\( f(x) \)[/tex]:

[tex]\[
71 = 16 \cdot 16^{\frac{1}{2}} + C
\][/tex]

Simplify this equation:
- Calculate [tex]\( 16^{\frac{1}{2}} \)[/tex], which is 4.
- So, [tex]\( 16 \times 4 = 64 \)[/tex].

[tex]\[
71 = 64 + C
\][/tex]

Solve for [tex]\( C \)[/tex]:
[tex]\[
C = 71 - 64 = 7
\][/tex]

3. Write the final function [tex]\( f(x) \)[/tex]:

Substitute [tex]\( C = 7 \)[/tex] back into the expression for [tex]\( f(x) \)[/tex]:

[tex]\[
f(x) = 16x^{\frac{1}{2}} + 7
\][/tex]

So, the function [tex]\( f(x) \)[/tex] is [tex]\( f(x) = 16\sqrt{x} + 7 \)[/tex].

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Rewritten by : Barada