DETAILS $$ ] $$ MY NOTES Find all relative extrema. Use the second derivative test where applicable. (If an answer does not exist, enter DNE.) $$ f(x)=(5 x-9)^{2} $$ relative maximum $$ \quad(x, y)=(\square) $$ relative minimurn $$ \quad(x, y)=(\square) $$

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**Step 1. Identify the function and compute its derivative**

The given function is
$$
f(x) = (5x - 9)^2.
$$
To find relative extrema, we first compute the derivative $$ f'(x) $$. Using the chain rule,
$$
f'(x) = 2(5x - 9) \cdot 5 = 10(5x - 9).
$$

**Step 2. Find critical points**

We set the derivative equal to zero to find the critical points:
$$
10(5x - 9) = 0.
$$
Dividing both sides by 10,
$$
5x - 9 = 0.
$$
Solving for $$ x $$,
$$
x = \frac{9}{5}.
$$

**Step 3. Use the second derivative test**

We now compute the second derivative $$ f''(x) $$ to determine the nature of the critical point. Differentiate $$ f'(x) $$:
$$
f''(x) = \frac{d}{dx}[10(5x - 9)] = 10 \cdot 5 = 50.
$$
Since
$$
f''\left(\frac{9}{5}\right) = 50 > 0,
$$
the function has a local minimum at $$ x = \frac{9}{5} $$.

**Step 4. Compute the minimum point**

Substitute $$ x = \frac{9}{5} $$ into $$ f(x) $$:
$$
f\left(\frac{9}{5}\right) = \left(5 \cdot \frac{9}{5} - 9\right)^2 = (9 - 9)^2 = 0.
$$
Thus, the relative minimum is at
$$
\left(\frac{9}{5}, 0\right).
$$

**Step 5. Determine the relative maximum**

Since $$ f(x)=(5x-9)^2 $$ is a quadratic function that opens upward (its second derivative is positive everywhere), it does not have a relative maximum. Therefore, the relative maximum is $$ \text{DNE} $$.

**Final Answers**

Relative maximum $$ \quad (x, y) = \text{DNE} $$  
Relative minimum $$ \quad (x, y) = \left(\frac{9}{5}, 0\right) $$
Relative maximum: DNE Relative minimum: $$ \left(\frac{9}{5}, 0\right) $$

DETAILS MY NOTES
Find all relative extrema. Use the second derivative test where applicable. (If an answer does not exist, enter DNE.)

relative maximum
relative minimurn

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DETAILS MY NOTES Find all relative extrema. Use the second derivative test where applicable. (If an answer does not exist, enter DNE.) relative maximum relative minimurn