Let f(x) = x^3 - 12r^2 + 45x + 14. Find me open Intervals on which f is increasi
ID: 2862756 • Letter: L
Question
Let f(x) = x^3 - 12r^2 + 45x + 14. Find me open Intervals on which f is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima). f is increasing on the intervals f is decreasing on the intervals The relative maxima of f occur at x = The relative minima of f occur at x = In the first two, your answer should either be a single interval, such as (0.1), a comma separated list of intervals. such as (-inf, 2). (3, 4). or the word "none". In the last two your answer should be a comma separated lot of x values or the word "none".Explanation / Answer
f(x)=x3-12x2+45x +14
differentiate with respect to x
f '(x)=3x2-24x+45
1)f is increasing when f '(x)>0
=>3x2-24x+45>0
=>3(x2-8x+15)>0
=>(x-3)(x-5)>0
=>x<3 ,x>5
x=(-inf,3)U(5,inf)
2)
f is increasing when f '(x)<0
=>3x2-24x+45<0
=>3(x2-8x+15)<0
=>(x-3)(x-5)<0
=>x>3 ,x<5
x=(3,5)
first derivative test :
3)at x =3 ,f '(x) changes from positive to negative => relative maximum at x =3
4)at x =5 ,f '(x) changes from negative to positive => relative minimum at x =5
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