a)
f′(x) is {positivenegative} when f(x) is {strictly increasingstrictly decreasing}.
⟹f(x) is strictly increasing on the interval (−∞,1)∪(3,∞) and strictly decreasing on the interval (1,3).
What about when x = 1 or x = 3?
Let ε be an arbitrary constant, infinitesmally small.
f′(1−ε)>0 and f′(1+ε)<0
At x = 1, local maximum of f(x) occurs.
f′(3−ε)<0 and f′(3+ε)>0
At x = 3, local minimum of f(x) occurs.
f′(x) is {increasingdecreasingconstant} when f(x) {is convexis concavehas a point of inflexion}.
⟹f(x) is concave on the interval (−∞,2) and is convex on the interval (2,∞).
Point of inflexion of f(x) occurs at x=2.
b)
f′(x) is {increasingdecreasingconstant} when f″(x) is {positivenegative0}.
⟹f″(x) is negative on the interval (−∞,2) and is positive on the interval (2,∞).
f″(2)=0
.