1. **Problem Statement:** Consider the function $$P(x) = \frac{x^2}{x^2 + 4}$$. We will analyze its domain, range, first and second derivatives, intervals of increase/decrease, concavity, inflection points, and asymptotes. 2. **Domain and Range:** - The denominator $$x^2 + 4$$ is always positive since $$x^2 \geq 0$$ and 4 is positive. - Therefore, the domain is all real numbers: $$(-\infty, \infty)$$. - For the range, note that $$P(x) = \frac{x^2}{x^2 + 4}$$ is always between 0 and 1 because $$x^2 \geq 0$$ and $$x^2 + 4 > x^2$$. - As $$x \to \pm \infty$$, $$P(x) \to 1$$. - At $$x=0$$, $$P(0) = 0$$. - Hence, the range is $$[0, 1)$$. 3. **First Derivative and Intervals of Increase/Decrease:** - Use the quotient rule: $$P'(x) = \frac{(2x)(x^2 + 4) - x^2(2x)}{(x^2 + 4)^2}$$. - Simplify numerator: $$2x(x^2 + 4) - 2x^3 = 2x^3 + 8x - 2x^3 = 8x$$. - So, $$P'(x) = \frac{8x}{(x^2 + 4)^2}$$. - Critical point at $$x=0$$. - For $$x < 0$$, $$P'(x) < 0$$ (decreasing). - For $$x > 0$$, $$P'(x) > 0$$ (increasing). 4. **Second Derivative and Concavity/Inflection Points:** - Differentiate $$P'(x)$$ using quotient rule: $$P''(x) = \frac{8(x^2 + 4)^2 - 8x \cdot 2(x^2 + 4)(2x)}{(x^2 + 4)^4}$$. - Simplify numerator: $$8(x^2 + 4)^2 - 32x^2(x^2 + 4) = 8(x^2 + 4)[(x^2 + 4) - 4x^2] = 8(x^2 + 4)(4 - 3x^2)$$. - So, $$P''(x) = \frac{8(x^2 + 4)(4 - 3x^2)}{(x^2 + 4)^4} = \frac{8(4 - 3x^2)}{(x^2 + 4)^3}$$. - Set numerator zero for inflection points: $$4 - 3x^2 = 0 \Rightarrow x = \pm \frac{2}{\sqrt{3}}$$. - Concave up where $$P''(x) > 0$$: when $$4 - 3x^2 > 0 \Rightarrow |x| < \frac{2}{\sqrt{3}}$$. - Concave down where $$|x| > \frac{2}{\sqrt{3}}$$. 5. **Asymptotes:** - Vertical asymptotes occur where denominator is zero; here, $$x^2 + 4 \neq 0$$ for all real $$x$$, so no vertical asymptotes. - Horizontal asymptote as $$x \to \pm \infty$$: $$\lim_{x \to \pm \infty} P(x) = \lim_{x \to \pm \infty} \frac{x^2}{x^2 + 4} = 1$$. - So, horizontal asymptote at $$y = 1$$. **Final answers:** - Domain: $$(-\infty, \infty)$$ - Range: $$[0, 1)$$ - Increasing on $$(0, \infty)$$, decreasing on $$(-\infty, 0)$$ - Concave up on $$\left(-\frac{2}{\sqrt{3}}, \frac{2}{\sqrt{3}}\right)$$, concave down elsewhere - Inflection points at $$x = \pm \frac{2}{\sqrt{3}}$$ - Horizontal asymptote at $$y=1$$, no vertical asymptotes This completes the curve sketching analysis.