To find the minimum value of f(x)=x3+(10−x)3, the derivative is needed.
The derivative of x3 is 3x2
The derivative of (10−x)3 is 3(10−x)2×(−1) by the chain rule.
f′(x)=3x2−3(10−x)2
At the minimum point, the derivative equals zero:
3x2−3(10−x)2=0
3[x2−(10−x)2]=0
x2−(10−x)2=0
x2=(10−x)2
When two values squared are equal, the values are equal or opposite:
x=10−x
x+x=10
2x=10
x=5
The minimum value occurs at x=5.
At this point, both terms x3 and (10−x)3 are equal, creating symmetry: f(5)=53+53=125+125=250