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${:[f^(')(x)=12e^(x)" and "f(4)=-16+12e^(4)] ,Find [f(0)=]:}$

$f'(x)$ = $12e^x$ and $f(4)$ = $-16 + 12e^4$ . Find $f(0)$ .

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Evaluate integers raised to rational exponents

Full solution

Q.

f'(x)

=

12e^x

and

f(4)

=

-16 + 12e^4

. Find

f(0)

.

Identify derivative and function value: Identify the derivative and the function value at $x=4$ . $f'(x) = 12e^x$ , $f(4) = -16 + 12e^4$
Use derivative to find general form: Use the derivative to find the general form of $f(x)$ . Since $f'(x) = 12e^x$ , integrate to find $f(x)$ . $f(x) = 12e^x + C$ , where $C$ is the constant of integration.
Find constant using given value: Use the given $f(4)$ to find the constant $C$ . $\newline$ $f(4) = -16 + 12e^4 = 12e^4 + C$ $\newline$ Solve for $C$ : $C = -16 + 12e^4 - 12e^4 = -16$
Write complete function: Write the complete function $f(x)$ . $f(x) = 12e^x - 16$
Calculate $f(0)$ : Calculate $f(0)$ using the complete function. $\newline$ $f(0) = 12e^0 - 16 = 12(1) - 16 = -4$

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