How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x)= 3x^4+4x^3-12x^2+5#?

Question

Ezekiel Alexander · Accepted Answer

Tthe function has a maximum at #x=0#
The function has a minimum at #x=1#
The function has a minimum at #x=2# Given - #f(x)= 3x^4+4x^3-12x^2+5# find the first two derivatives #f^'=12x^3+12x^2-24x# #f^('')=36x^2+24x-24# Set first derivative equal to zero #f^' = 0=>12x^3+12x^2-24x=0# Find the values of #x# #12x(x^2+x-2)# #12x(x^2+2x-x-2)# #12x[x(x+2)-1(x+2)]# #12x(x-1)(x+2)# #12x=0# #x=0# #x-1=0# #x=1# #x+2=0# #x=2# #x # has three values At #x=0# #f^('')=36(0)^2+24(0)-24=-24 < 0# At #x=0; f^'=0; f^('')<0# Hence the function has a maximum at #x=0# At #x=1 # # f^('')=36(1)^2+24(1)-24# # f^('')=36+24-24=36 > 0# At #x=0; f^'=0; f^('')<>0# Hence the function has a minimum at #x=1# At #x=2# # f^('')=36(2)^2+24(2)-24# # f^('')=144+48-24=168 > 0# At #x=0; f^'=0; f^('')<>0# Hence the function has a minimum at #x=2# graph{3x^4+4x^3-12x^2+5 [-10, 10, -5, 5]} #

Isabella Ackerman · Answer

undefined To determine where the function is increasing or decreasing, and where relative maxima and minima occur for $ f(x) = 3x^4 + 4x^3 - 12x^2 + 5 $: 1. Find the first derivative of the function. 2. Set the first derivative equal to zero to find critical points. 3. Use the first derivative test to determine the intervals where the function is increasing or decreasing. 4. Find the second derivative of the function. 5. Use the second derivative test to determine where the relative maxima and minima occur. First Derivative: $ f'(x) = 12x^3 + 12x^2 - 24x $ Critical Points: Set $ f'(x) = 0 $ and solve for $ x $. Second Derivative: $ f''(x) = 36x^2 + 24x - 24 $ Use the Second Derivative Test: - If $ f''(x) > 0 $, the function is concave up, indicating a relative minimum. - If $ f''(x) < 0 $, the function is concave down, indicating a relative maximum. Determine intervals of increase and decrease using the First Derivative Test: - If $ f'(x) > 0 $, the function is increasing. - If $ f'(x) < 0 $, the function is decreasing. Combine the information from steps 3 and 5 to locate relative maxima and minima.