1. **Problem Statement:** Find the critical values, intervals of increase and decrease, and relative maxima and minima for the cost function $$C(x) = -0.1x^2 + 18x - 340$$ on the domain $$0 < x < 400$$. 2. **Formula and Rules:** - Critical values occur where the derivative $$C'(x)$$ is zero or undefined. - The function is increasing where $$C'(x) > 0$$ and decreasing where $$C'(x) < 0$$. - Relative maxima occur at critical points where the function changes from increasing to decreasing. - Relative minima occur at critical points where the function changes from decreasing to increasing. 3. **Find the derivative:** $$C'(x) = \frac{d}{dx}(-0.1x^2 + 18x - 340) = -0.2x + 18$$ 4. **Find critical values by setting derivative to zero:** $$-0.2x + 18 = 0$$ $$-0.2x = -18$$ $$x = \frac{-18}{-0.2} = 90$$ 5. **Determine intervals of increase and decrease:** - For $$x < 90$$, pick $$x=0$$: $$C'(0) = -0.2(0) + 18 = 18 > 0$$, so $$C(x)$$ is increasing on $$(0,90)$$. - For $$x > 90$$, pick $$x=100$$: $$C'(100) = -0.2(100) + 18 = -20 + 18 = -2 < 0$$, so $$C(x)$$ is decreasing on $$(90,400)$$. 6. **Identify relative maxima and minima:** - At $$x=90$$, the function changes from increasing to decreasing, so there is a relative maximum at $$x=90$$. - There is no point where the function changes from decreasing to increasing, so there are no relative minima (DNE). **Final answers:** - Critical value: $$x=90$$ - Increasing interval: $$(0,90)$$ - Decreasing interval: $$(90,400)$$ - Relative maximum at $$x=90$$ - No relative minimum (DNE)