Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). f(x)=x^{3}-6x^{2} text{on} [-1,5]

sanuluy 2020-11-01 Answered
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). f(x)=x36x2on[1,5]
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Dora
Answered 2020-11-02 Author has 98 answers

Step 1 Let us find the critical points , absolute maximum and minimum of given function Given: f(x)=x36x2on[1,5]

Step 2 f(x)=x36x2
f(x)=3x212x
0=3x(x4)
3x=0,x4=0
x=0,x=4 Critical point =0,4

Step 3 To find absolute maximum and absolute minimum: f(x)=x36x2
x=1
f(1)=(1)36(1)2
f(1)=16
f(1)=7
x=0
f(0)=06(0)
f(0)=0
x=4
f(4)=436(4)2
f(4)=646(16)
f(4)=6496
f(4)=32
x=5
f(5)=536(5)2
f(5)=125150
f(5)=25 Absolute maximum at y=0, x=0 Absolute minimum at y=-32, x=5

Step 4 Answer: Critical points at x=0,4 Absolute maximum at x=0 Absolute minimum at x=5

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