Find minimum and maximum values of the functioin f(x) = (3-x)**2 + (1+x)**3 for 0<=x<=2
f(x)=(3−x)2+(1+x)3f′(x)=2⋅(3−x)⋅(−1)+3⋅(1+x)2⋅(1)f′(x)=3x2+8x−3f″(x)=3⋅2⋅x+8f″(x)=6x+8
minimum and maximum: f′(x)=0
3x2+8x−3=0 x=−b±√b2−4ac2a x1,2=−8±√82−4⋅3⋅(−3)2⋅3x1,2=−8±√64+366x1,2=−8±√1006x1,2=−8±106x1=−8+106x1=26x1=13y″(13)=6⋅13+8=10>0→Minimumx2=−8−106x2=−186x2=−3y″(13)=6⋅(−3)+8=−10<0→Maximum
0≤x≤2 there is a Minimum at x=13
