Chapter 4.2, Problem 17E

a)

(i)

To determine

Sketch the Zeros of polynomial $y=x^2-4$ on a line along with the Zeros of its first derivative.

(ii)

To determine

Sketch the Zeros of polynomial $y=x^2+8x+15$ on a line along with the Zeros of its first derivative.

(iii)

To determine

Sketch the Zeros of polynomial $y=x^3-3x^2+4=\left(x+1\right){\left(x-2\right)}^2$ on a line along with the Zeros of its first derivative.

(iv)

To determine

Sketch the Zeros of polynomial $y=x^3-33x^2+216x=x\left(x-9\right)\left(x-24\right)$ on a line along with the Zeros of its first derivative.

b)

To determine

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 $n{x}^{n-1}+\left(n-1\right)a_{n-1}{x}^{n-2}+\cdots +a_1$.