Question

Let f(x)= max( 2x + 1, 3 – 4x), where x is any real number. then, the minimum possible value of f(x) is

Answer 1

The function ff is convex and its subdifferential is given by

∂f(x)=⎧⎩⎨⎪⎪⎪⎪{−4},[−4,2],{2},x<13,x=13,x>13.∂f(x)={{−4},x<13,[−4,2],x=13,{2},x>13.

Since ff is convex, then x^x^ minimizes ff iff 0∈∂f(x^)0∈∂f(x^). It follows that the minimizing x^x^ is 1313, and hence the minimum value of ffis 5353.
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