Calculus 1-Maximum and Minimum Values, guaranteed 100% Pass

1


Maximum and Minimum Values



Def. A function 𝑓(𝑥) has an Absolute Maximum (or Global Maximum) at 𝑥 = 𝑐 if

𝑓(𝑐) ≥ 𝑓(𝑥) for all 𝑥 in the domain of 𝑓(𝑥). 𝑓(𝑐) is called the Maximum Value
of 𝑓(𝑥) for all 𝑥 in the domain of 𝑓(𝑥).

A function 𝑓(𝑥) has an Absolute Minimum (or Global Minimum) at 𝑥 = 𝑐 if

𝑓(𝑐) ≤ 𝑓(𝑥) for all 𝑥 in the domain of 𝑓(𝑥). 𝑓(𝑐) is called the Minimum Value
of 𝑓(𝑥) for all 𝑥 in the domain of 𝑓(𝑥).

The Maximum and Minimum Values of 𝑓(𝑥) are called the Extreme Values of
𝑓(𝑥).




𝑓(𝑥) has a global maximum at 𝑥 = 4.
𝑓(𝑥) has a global minimum at 𝑥 = 2.




Def. A function 𝑓(𝑥) has a Local Maximum (or Relative Maximum) at 𝑥 = 𝑐 if

𝑓(𝑐) ≥ 𝑓(𝑥) when 𝑥 is near 𝑐.

A function 𝑓(𝑥) has a Local Minimum (or Relative Minimum) at 𝑥 = 𝑐 if

𝑓(𝑐) ≤ 𝑓(𝑥) when 𝑥 is near 𝑐.

(Note: 𝑥 = 𝑐 cannot be an endpoint).

, 2


Ex. 𝑓 (𝑥 ) = 𝑥 2 for −∞ < 𝑥 < ∞ has a local and global minimum at 𝑥 = 0,

but no global or local maximum.




𝑓(𝑥) = 𝑥 2




−3 −2 −1 0 1 2 3


Ex. 𝑓(𝑥 ) = 9 − 𝑥 2 for −3 ≤ 𝑥 ≤ 3 has a local and global maximum at
(0, 9) and global (but not local) minima at (−3, 0) and (3, 0).
(0,9) Maximum (Global and Local)




𝑦 = 9 − 𝑥2




(−3,0) Global Minimum (3,0) Global Minimum