Let f(x) be a continuous function on I=[a,b], and let g(x) be a differentiable function on I. Let g(a)=0 and c≠0 a constant. Prove that if

|g(x) f(x)+c g′(x)|≤|g(x)| for all x∈I,

then g(x)=0 for all x∈I.