Home/Differential Calculus/Optimization/Criteria on closed intervalsSection 9.2Criteria on closed intervalsFormulaCopy LaTeXQmax=max{Q(a),Q(b),Q(c1),…,Q(ck)}Q_{\max}=\max\{Q(a),Q(b),Q(c_1),\ldots,Q(c_k)\}Qmax=max{Q(a),Q(b),Q(c1),…,Q(ck)}Absolute maximum on [a,b][a,b][a,b] by comparing critical points cic_ici and endpoints.View detail →FormulavizCopy LaTeXA=xy,2x+2y=P⇒A(x)=x(P2−x)A=xy,\quad 2x+2y=P \Rightarrow A(x)=x\left(\frac{P}{2}-x\right)A=xy,2x+2y=P⇒A(x)=x(2P−x)Typical example: maximum area with fixed perimeter (rectangle → square). Here PPP is the full perimeter (P=2x+2yP=2x+2yP=2x+2y); the semiperimeter is C=P/2=x+yC=P/2=x+yC=P/2=x+y.2x+2y=P⇒A(x)=x(P2−x)2x+2y=P \Rightarrow A(x)=x\left(\frac{P}{2}-x\right)2x+2y=P⇒A(x)=x(2P−x)View detail →FormulavizCopy LaTeXV=πr2h,S=2πr2+2πrh=C⇒V(r)=C2r−πr3V=\pi r^2h,\quad S=2\pi r^2+2\pi rh=C \Rightarrow V(r)=\frac{C}{2}r-\pi r^3V=πr2h,S=2πr2+2πrh=C⇒V(r)=2Cr−πr3Typical example: cylinder with fixed surface area and maximum volume.S=2πr2+2πrh=C⇒V(r)=C2r−πr3S=2\pi r^2+2\pi rh=C \Rightarrow V(r)=\frac{C}{2}r-\pi r^3S=2πr2+2πrh=C⇒V(r)=2Cr−πr3View detail →
FormulaCopy LaTeXQmax=max{Q(a),Q(b),Q(c1),…,Q(ck)}Q_{\max}=\max\{Q(a),Q(b),Q(c_1),\ldots,Q(c_k)\}Qmax=max{Q(a),Q(b),Q(c1),…,Q(ck)}Absolute maximum on [a,b][a,b][a,b] by comparing critical points cic_ici and endpoints.View detail →
FormulavizCopy LaTeXA=xy,2x+2y=P⇒A(x)=x(P2−x)A=xy,\quad 2x+2y=P \Rightarrow A(x)=x\left(\frac{P}{2}-x\right)A=xy,2x+2y=P⇒A(x)=x(2P−x)Typical example: maximum area with fixed perimeter (rectangle → square). Here PPP is the full perimeter (P=2x+2yP=2x+2yP=2x+2y); the semiperimeter is C=P/2=x+yC=P/2=x+yC=P/2=x+y.2x+2y=P⇒A(x)=x(P2−x)2x+2y=P \Rightarrow A(x)=x\left(\frac{P}{2}-x\right)2x+2y=P⇒A(x)=x(2P−x)View detail →
FormulavizCopy LaTeXV=πr2h,S=2πr2+2πrh=C⇒V(r)=C2r−πr3V=\pi r^2h,\quad S=2\pi r^2+2\pi rh=C \Rightarrow V(r)=\frac{C}{2}r-\pi r^3V=πr2h,S=2πr2+2πrh=C⇒V(r)=2Cr−πr3Typical example: cylinder with fixed surface area and maximum volume.S=2πr2+2πrh=C⇒V(r)=C2r−πr3S=2\pi r^2+2\pi rh=C \Rightarrow V(r)=\frac{C}{2}r-\pi r^3S=2πr2+2πrh=C⇒V(r)=2Cr−πr3View detail →