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Local extrema or distinct roots?

Source: ISI Entrance 2014, P3

May 11, 2014
real analysisreal analysis unsolved

Problem Statement

Consider f(x)=x4+ax3+bx2+cx+d  (a,b,c,d∈R)f(x)=x^4+ax^3+bx^2+cx+d\; (a,b,c,d\in\mathbb{R}). It is known that ff intersects X-axis in at least 33 (distinct) points. Show either ff has 44 distinct\mathbf{distinct} real roots or it has 33 distinct\mathbf{distinct} real roots and one of them is a point of local maxima or minima.
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