Find the absolute (global) maximum [math: y] value of the function:[math: f(x)=2x3+0x2+−24x+5]On the interval [math: [−9,10]]

Question

BlackTom AI · Accepted Answer

To tackle this optimization on a closed interval, I’ll start by locating critical points where the derivative is zero or undefined.

First, compute the derivative: f'(x) = d/dx [2x^3 + 0x^2 - 24x + 5] = 6x^2 - 24 = 6(x^2 - 4) = 6(x - 2)(x + 2).
Setting f'(x) = 0 gives x = -2 and x = 2 as critical points within the interval [-9, 10].

Next, evaluate the function at the endpoints and at the critical points:
- At x = -9: f(-9) = 2(-9)^3 - 24(-9) + 5 = 2(-729) + 216 + 5 = -1458 + 221 = -1237.
- At x = -2: f(-2) = 2(-2)^3 - 24(-2) + 5 = 2(-8) + 48 + 5 = -16 + 53 = 37.
- At x = 2: f(2) = 2(2)^3 - 24(2) + 5 = 2(8) - 48 + 5 = 16 - 48 + 5 = -27.
- At x = 10: f(10) = 2(10)^3 - 24(10) + 5 = 2(1000) - 240 + 5 = 2000 - 240 + 5 = 1765.

Now compare these values to determine the absolute maximum on the interval. The candidate values are -1237, 37, -27, and 1765.
Among them, the largest is 1765, which occurs at x = 10. Therefore, the absolute (global) maximum y-value on [-9, 10] is 1765.

UCIC 202503 EMTH118 Quiz 10

Find the absolute (global) maximum [math: y] value of the function:[math: f(x)=2x3+0x2+−24x+5]On the interval [math: [−9,10]]

查看解析

登录即可查看完整答案

类似问题

Find the absolute (global) maximum \( y \) value of the function:\( f(x)=2x^3+0x^2+-24x+5 \)On the interval \( [-9, 10] \)

Maxima and Minima Graph the function Find the value of at which the given has a local maximum. Round to the nearest hundredth. [Fill in the blank] Find the value of at which the given has a local minimum. Round to the nearest hundredth. [Fill in the blank]

Find the extreme values of the function and where they occur. y =

Question at position 8 On the interval [-1,+1], the function y = 4 + x2 -x3 has an absolute maximum when x =-0.50.51.-1.0.

What is the absolute maximum value of f ( x ) = x 2 x − 7 3 {"version":"1.1","math":"f(x)=x^2\sqrt[3]{x-7}"} on [ − 4 , 8 ] ? {"version":"1.1","math":"[-4,8]?"}

For the function f \colon \mathbb{R} \rightarrow \mathbb{R}, \quad f(x) = (4-x)e^x, consider its global maximum. This global maximum occurs at which value of x?

Find the absolute minimum and absolute maximum values of 𝑓 ( 𝑥 ) = 𝑥 3 − 3 𝑥 + 2 on the closed interval [ − 2 , 3 ] .

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单