Flightpath of Hang Glider

Calculus Level 3

The initial flightpath of a hang glider can be described as \[ f(x) = x^4 - \frac{28}{3}x^3 + 2ax^2, \] where \(a\) depends on certain physical properties of the hang glider and other environmental conditions. To avoid significant turbulence which may destabilize the hang glider, the flight path should not have local maxima. How many positive integers \(a\ (<1000)\) are there such that \(f(x)\) has no local maxima?

Image credit: The Suitcase Scholar

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