Let f(x)=-x3+9x2-15x+ (a) Find the intervals where f is increasing and where f i

Question

Let f(x)=-x3+9x2-15x+ (a) Find the intervals where f is increasing and where f is decreasing. (b) Find intervals where f is concave upward and where f is concave downward. (c) Find the local minimum and maximum values. (d) Find inflection points (e) Graph f Let f(x)=-x3+9x2-15x+ (a) Find the intervals where f is increasing and where f is decreasing. (b) Find intervals where f is concave upward and where f is concave downward. (c) Find the local minimum and maximum values. (d) Find inflection points (e) Graph f

Explanation / Answer

you had the question posted as f(x)=-x3+9x2-15x+, not sure of there should be something after the -15x term. i am assuming there wasnt.

f(x)=-x3+9x2-15x
f'(x)=-3x2+18x-15       =-3(x2-6x+5)       =-3(x-5)(x-1) when f'(x)=0 => x=5 and x=1 are the critical points now, seperate the real line in 3 intervals, (-,1) (1,5) (5,) pick one number fron each of the intercal, say fron the (-,1) intercal i pick -1 f(-1)= -(-1)3+9(-1)2-15(-1)=1+9+15=25>0, so f is increasing in this interval. for (1,5) , i pick 2 be my point f(2)=-8+36-30=-2<0 so f is decreasing for (5,), say i pick 6 as my point to check f(6)=-216+324-90=18>0 so f is increasing again. b). take the second derivative, f''(x)=-6x+18

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