1. **State the problem:** We need to find the minimum and maximum values of the function $$f(x) = 10x^2 - \lg x$$ for $$1 \leq x \leq 1000$$. 2. **Recall the function and domain:** The function is $$f(x) = 10x^2 - \lg x$$ where $$\lg x$$ is the base-10 logarithm of $$x$$. 3. **Find the derivative to locate critical points:** $$f'(x) = \frac{d}{dx}(10x^2) - \frac{d}{dx}(\lg x) = 20x - \frac{1}{x \ln 10}$$ 4. **Set the derivative equal to zero to find critical points:** $$20x - \frac{1}{x \ln 10} = 0$$ Multiply both sides by $$x$$: $$20x^2 = \frac{1}{\ln 10}$$ Solve for $$x^2$$: $$x^2 = \frac{1}{20 \ln 10}$$ Therefore, $$x = \sqrt{\frac{1}{20 \ln 10}}$$ 5. **Calculate the approximate value of the critical point:** Since $$\ln 10 \approx 2.302585$$, $$x \approx \sqrt{\frac{1}{20 \times 2.302585}} = \sqrt{\frac{1}{46.0517}} \approx \sqrt{0.0217} \approx 0.147$$ 6. **Check if the critical point is within the domain:** The domain is $$1 \leq x \leq 1000$$, but $$0.147 < 1$$, so the critical point is outside the domain. 7. **Evaluate the function at the domain endpoints:** - At $$x=1$$: $$f(1) = 10(1)^2 - \lg 1 = 10 - 0 = 10$$ - At $$x=1000$$: $$f(1000) = 10(1000)^2 - \lg 1000 = 10 \times 1,000,000 - 3 = 10,000,000 - 3 = 9,999,997$$ 8. **Determine minimum and maximum:** Since the critical point is outside the domain, the minimum and maximum occur at the endpoints. - Minimum value is $$f(1) = 10$$ - Maximum value is $$f(1000) = 9,999,997$$