10-point analysis of the Polynomial Function f(x) = 4x^4 - 17x^2 + 4

f(x) = 4x^4 - 17x^2 + 4
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  1. It factorises: f(x)=(x2-4)(4x2-1)=(x+2)(x-2)(2x+1)(2x-1)
  2. There are 4 zeroes (roots): -2, 2, -½, ½. 
  3. -2, 2, -½, ½ are also x-intercepts.
  4. All zeroes are real (not complex).
  5. f(x) is an even function because f(x)=f(-x).
  6. df/dx=16x3-34x=0 at extrema. 2x(8x2-17)=0, so x=0, ±√(17/8). y values are 4, -225/16, -225/16.
  7. Maximum at (0,4).
  8. Minimum at (-√(17/8),-225/16).
  9. Minimum at (√(17/8),-225/16).
  10. y-intercept at (0,4), coinciding with maximum.
by Top Rated User (1.2m points)

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