Consider the function f(x) = x^3 - 9x^2 + 24x + 1. a. Find the critical numbers
ID: 2877425 • Letter: C
Question
Consider the function f(x) = x^3 - 9x^2 + 24x + 1. a. Find the critical numbers of f(x) if any. b. Find the open intervals on which f(x) is increasing or decreasing. c. Use the First Derivative Test to find any relative extrema for the function Find the absolute extreme for the function f(x) = 2sin(x) - x on the closed interval [0, 2 pi]. Use the Second Derivative Test to find any relative extrema for the function f(x) = x^3 - 3x^2 - 24x + 2 Find the following limit lim_x rightarrow (-infinity) 9 - 5x/Squareroot 25x^2 + 2 A rectangular solid with a square base has a surface area of 657 square centimeters. Find the dimensions that will result in a solid with maximum volume. Round your answers to two decimal places.Explanation / Answer
1)
a)
f(x) = x^3 -9x^2 +24x +1
f'(x) = 3x^2 -18x+24
put f'(x) = 0
3x^2 -18x+24 = 0
x^2 - 6x + 8 = 0
(x-4)(x-2)=0
x=4 and x=2 are critical numbers
b)
for x<2, f'(x) is positive. so f(x) is increasing
for x between 2 and 4, f'(x) is negative. so, f(x) is decreasing
for x>4, f'(x) is positive. so f(x) is increasing
f(x) is increasing on
(-inf,2)U(4,inf)
f(x) is decreasing on
(-2,4)
c)
f(x) is increasing on
(-inf,2)U(4,inf)
f(x) is decreasing on
(-2,4)
clearly, x=2 is maxima
x=4 is minima
I am allowed to answer only 1 question at a time