What is olic diffoicnce betwsen the maximum value of \( y=-x^{4}-2 x^{3}+5 \) gnd the minimum value of \( y=x^{4}+x-4 \) ? (A) 1 (B) 1.5 (C) 2.7 (D) 9 (iD) 10.7

To find the difference between the maximum value of \( y = -x^4 - 2x^3 + 5 \) and the minimum value of \( y = x^4 + x - 4 \), we will first analyze each function separately. ### Step 1: Find the maximum value of \( y = -x^4 - 2x^3 + 5 \) 1. **Find the derivative**: \[ y' = -4x^3 - 6x^2 \] Factor out common terms: \[ y' = -2x^2(2x + 3) \] 2. **Set the derivative to zero**: \[ -2x^2(2x + 3) = 0 \] This gives us: \[ x^2 = 0 \quad \Rightarrow \quad x = 0 \] \[ 2x + 3 = 0 \quad \Rightarrow \quad x = -\frac{3}{2} \] 3. **Evaluate the function at critical points**: - At \( x = 0 \): \[ y(0) = -0^4 - 2(0)^3 + 5 = 5 \] - At \( x = -\frac{3}{2} \): \[ y\left(-\frac{3}{2}\right) = -\left(-\frac{3}{2}\right)^4 - 2\left(-\frac{3}{2}\right)^3 + 5 \] Calculate each term: \[ -\left(-\frac{3}{2}\right)^4 = -\frac{81}{16}, \quad -2\left(-\frac{3}{2}\right)^3 = -2\left(-\frac{27}{8}\right) = \frac{27}{4} = \frac{108}{16} \] Thus, \[ y\left(-\frac{3}{2}\right) = -\frac{81}{16} + \frac{108}{16} + 5 = \frac{27}{16} + 5 = \frac{27}{16} + \frac{80}{16} = \frac{107}{16} \approx 6.6875 \] 4. **Maximum value**: The maximum value of \( y = -x^4 - 2x^3 + 5 \) is \( 6.6875 \). ### Step 2: Find the minimum value of \( y = x^4 + x - 4 \) 1. **Find the derivative**: \[ y' = 4x^3 + 1 \] 2. **Set the derivative to zero**: \[ 4x^3 + 1 = 0 \quad \Rightarrow \quad 4x^3 = -1 \quad \Rightarrow \quad x^3 = -\frac{1}{4} \quad \Rightarrow \quad x = -\frac{1}{\sqrt[3]{4}} \approx -0.63 \] 3. **Evaluate the function at the critical point**: \[ y\left(-\frac{1}{\sqrt[3]{4}}\right) = \left(-\frac{1}{\sqrt[3]{4}}\right)^4 + \left(-\frac{1}{\sqrt[3]{4}}\right) - 4 \] Calculate: \[ \left(-\frac{1}{\sqrt[3]{4}}\right)^4 = \frac{1}{\left(\sqrt[3]{4}\right)^4} = \frac{1}{4^{4/3}} = 4^{-4/3} \] \[ y\left(-\frac{1}{\sqrt[3]{4}}\right) \text{ is complex to calculate directly, so we can evaluate at } x = -2 \text{ and } x = 0. \] - At \( x = -2 \): \[ y(-2) = (-2)^4 + (-2) - 4 = 16 - 2 - 4 = 10 \] - At \( x = 0 \): \[ y(0) = 0^4 + 0 - 4 = -4 \] 4. **Minimum value**: The minimum value of \(