Find the interval increasing, decreasing, maximum, minimum value and or range and or domain for:?

Question

a. h(x)=8-|x+4| increasing=______________, decreasing= _____________,range= _________, maximum= ______________

b. f(x)=x^2-6x+14 increasing=______________, decreasing= ______________,domain= _________, minimum= ______________

c. m(x) = 14+21x-2x^2-3x^3 increasing=______________, decreasing= ______________,
minimum= ____________, maximum= ______________

Answer 1

a)

The absolute value of anything is always positive so h(x) cannot exceed 8, therefore 8 is the absolute maximum when x=-4. The domain is x∈(-∞,∞) and the range y∈(-∞,8], where y=h(x).

When x<-4, h(x) is increasing to the maximum, and when x>-4 it's decreasing from the maximum.

b) 

f(x)=x²-6x+14=x²-6x+9+5=(x-3)²+5.

(x-3)² is always positive, and f(x) is minimum when x=3 and f(x)=5.

When x<3 f(x) is decreasing towards the minimum and when x>3 it's increasing from the minimum.

Domain is the same as (a), but the range is y=f(x)∈[5,∞).

c)

m(x)=14+21x-2x²-3x³

m'(x)=21-4x-9x². m'(x)=0 at extrema, so x=(-4±√(16+756))/18⇒x=1.32138, -1.76583 approx.

m"(x)=-4-18x is <0 (maximum) when x>0 and is >0 (minimum) when x<0.

Therefore when x<-1.76583 m(x) is decreasing to the minimum, when -1.76583<x<1.32138 m(x) is increasing to the maximum, and when x>1.32138 it's decreasing from the maximum. The range is (-∞,∞) as is the domain.