If $$ f^{\prime}(x)=(2 x-3)^{2} $$ and $$ f(3)=11 $$, determine the equation of $$ f(x) $$

Question

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To find the equation of $$ f(x) $$, we need to integrate the derivative $$ f^{\prime}(x) = (2x - 3)^{2} $$ and then use the initial condition $$ f(3) = 11 $$ to determine the constant of integration.

### Step 1: Integrate $$ f^{\prime}(x) $$

We start by expanding $$ (2x - 3)^{2} $$:

$$
(2x - 3)^{2} = 4x^{2} - 12x + 9
$$

Now, we integrate $$ f^{\prime}(x) $$:

$$
f(x) = \int (4x^{2} - 12x + 9) \, dx
$$

Calculating the integral term by term:

1. The integral of $$ 4x^{2} $$ is $$ \frac{4}{3}x^{3} $$.
2. The integral of $$ -12x $$ is $$ -6x^{2} $$.
3. The integral of $$ 9 $$ is $$ 9x $$.

Putting it all together, we have:

$$
f(x) = \frac{4}{3}x^{3} - 6x^{2} + 9x + C
$$

where $$ C $$ is the constant of integration.

### Step 2: Use the initial condition to find $$ C $$

We know that $$ f(3) = 11 $$. We substitute $$ x = 3 $$ into the equation:

$$
f(3) = \frac{4}{3}(3)^{3} - 6(3)^{2} + 9(3) + C
$$

Calculating each term:

1. $$ \frac{4}{3}(3)^{3} = \frac{4}{3} \cdot 27 = 36 $$
2. $$ -6(3)^{2} = -6 \cdot 9 = -54 $$
3. $$ 9(3) = 27 $$

Now substituting these values into the equation:

$$
11 = 36 - 54 + 27 + C
$$

Simplifying the right side:

$$
11 = 36 - 54 + 27 + C \implies 11 = 9 + C
$$

Solving for $$ C $$:

$$
C = 11 - 9 = 2
$$

### Final Equation

Now we can write the final equation for $$ f(x) $$:

$$
f(x) = \frac{4}{3}x^{3} - 6x^{2} + 9x + 2
$$

Thus, the equation of $$ f(x) $$ is:

$$
\boxed{f(x) = \frac{4}{3}x^{3} - 6x^{2} + 9x + 2}
$$
The equation of $$ f(x) $$ is $$ f(x) = \frac{4}{3}x^{3} - 6x^{2} + 9x + 2 $$.

If and , determine the equation of

Upstudy AI Solution

Answer

Solution

Step 1: Integrate

Step 2: Use the initial condition to find

Final Equation

Beyond the Answer

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