Theorem 8.6 (Second Derivative Test)Suppose thatcis a critical pointof a functionf.1. Iff00(c)>0,fhas a relative minimum atc.2. Iff00(c)<0,fhas a relative maximum atc.3. Iff00(c) = 0,fmay have a relative maximum, minimum or neither ofthem.proof:Iff00(c)>0,f00(x)>0 whenxis close toc.Thus,f0is increasing over aninterval includingc. But we know thatf0(c) = 0. Therefore,f0(x)≤0 whenxis a bit less thancandf0(x)≥0 whenxis a bit greater thanc.fhas a relativeminimum atc. We may deal with the remaining case in whichf00(c)<0 in asimilar way.Example 8.7Find all the relative maximum and minimum of the functionf(x) = 2x3+ 3x2-12x+ 7.