Question

let (f(x)=2x^{3}+24x^{2}-54x + 19) be defined on (-10,10). find: a. the absolute maximum. write your answer as an ordered pair. b. find the absolute minimum. write your answer as an ordered pair.

Explanation:

Step1: Find the derivative of \(f(x)\)

Using the power - rule \((x^n)^\prime=nx^{n - 1}\), we have \(f^\prime(x)=6x^{2}+48x - 54\).

Step2: Set the derivative equal to zero and solve for \(x\)

\[

$$\begin{align*} 6x^{2}+48x - 54&=0\\ x^{2}+8x - 9&=0\\ (x + 9)(x - 1)&=0 \end{align*}$$

\]
So \(x=-9\) or \(x = 1\).

Step3: Evaluate \(f(x)\) at the critical points and endpoints

• Evaluate \(f(x)\) at \(x=-10\): \(f(-10)=2(-10)^{3}+24(-10)^{2}-54(-10)+19=-2000 + 2400+540 + 19=959\).
• Evaluate \(f(x)\) at \(x=-9\): \(f(-9)=2(-9)^{3}+24(-9)^{2}-54(-9)+19=2(-729)+24\times81 + 486+19=-1458+1944+486+19=991\).
• Evaluate \(f(x)\) at \(x = 1\): \(f(1)=2(1)^{3}+24(1)^{2}-54(1)+19=2 + 24-54 + 19=-9\).
• Evaluate \(f(x)\) at \(x = 10\): \(f(10)=2(10)^{3}+24(10)^{2}-54(10)+19=2000+2400-540 + 19=3879\).

Step4: Determine the absolute maximum and minimum

The largest value among \(f(-10),f(-9),f(1),f(10)\) is \(3879\) at \(x = 10\), and the smallest value is \(-9\) at \(x = 1\).

Answer:

a. \((10,3879)\)
b. \((1,-9)\)