15. a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative. i) \( y=x^{2}-4 \) ii) \( y=x^{2}+8 x+15 \) iii) \( y=x^{3}-3 x^{2}+4=(x+1)(x-2)^{2} \) iv) \( y=x^{3}-33 x^{2}+216 x=x(x-9)(x-24) \) b. Use Rolle's Theorem to prove that between every two zeros of \( x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} \) there lies a zero of \( n x^{n-1}+(n-1) a_{n-1} x^{n-2}+\cdots+a_{1} \)

### Part 15a: Plotting the Zeros #### i) \( y = x^2 - 4 \) - Zeros of \( y \): \( -2, 2 \) - Zero of \( y' \): \( 0 \) #### ii) \( y = x^2 + 8x + 15 \) - Zeros of \( y \): \( -3, -5 \) - Zero of \( y' \): \( -4 \) #### iii) \( y = x^3 - 3x^2 + 4 \) - Zeros of \( y \): \( -1, 2 \) - Zeros of \( y' \): \( 0, 2 \) #### iv) \( y = x^3 - 33x^2 + 216x \) - Zeros of \( y \): \( 0, 9, 24 \) - Zeros of \( y' \): \( 4, 18 \) ### Part 15b: Rolle's Theorem Rolle's Theorem states that if a function \( f \) is continuous on \([a, b]\) and differentiable on \((a, b)\), and \( f(a) = f(b) \), then there exists a \( c \) in \((a, b)\) such that \( f'(c) = 0 \). This proves that between every two zeros of \( f(x) \), there lies a zero of \( f'(x) \).