Extreme Values Test:

Consider a function ƒ be differential in a neighborhood of c, where ƒ has a great value at c. Then ƒ(c) is a maximum value if the sign of ƒ' modifies from plus to minus and ƒ(c) is a minimum value if the sign of ƒ' modifies from minus to plus as x passes through c.

Proof: Consider ƒ' changes sign from plus to minus as x passes during c. Then there exists a δ > 0 such that ƒ'(x) > 0 in ]c - δ, c[, and ƒ'(x) < 0 in ]c, c + δ[.

ƒ is sternly increasing in ]c - δ, c[, and ƒ is sternly decreasing in ]c, c + δ[. Consequently

ƒ'(x) < ƒ'(c) V x ?]c - δ, c[                               (1)

and, ƒ'(x) < ƒ'(c) V x ?]c, c + δ[                       (2)

Now, (1) and (2) mean that ƒ'(x) < ƒ'(c) V x ?]c - δ, c + δ[, x ≠ c

Thus ƒ has a maximum value at x = c. Same as it may be proved that ƒ has a minimum value at x = c if ƒ' modifies sign from minus to plus as x passes through c.