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Let g(x) = (x − r)2[(x − a)2 + b2], b>0, with A=(a,g(a)) and R=(r,0). When a≠r the tangent at A has x-intercept T=((a+r)/2,0). Assume (a − r)2 > 2b2, so the quartic has two inflection points. (i) Show that their x-coordinates satisfy…
Let g(x) = (x − r)2[(x − a)2 + b2], b>0, with A=(a,g(a)) and R=(r,0). When a≠r the tangent at A has x-intercept T=((a+r)/2,0). (i) Sketch the curve for a=r=−1 and b=2. (ii) For general a=r, state the coordinates of A and of the curve’s minimum point.
Prove that their sum is divisible by 3.
Prove that the sum of their squares is never divisible by 3.
A function g satisfies g′(x) = 4x3 − 6e−2x and g(0)=8. Find g(x).
Expand (z−u)(z−v)(z−w) and show that u+v+w=−q/r, uv+vw+wu=p/r, and uvw=−1/r.
You may use u+v+w=−q/r and uv+vw+wu=p/r. (i) Show that u2 + v2 + w2 = (q2 − 2pr)/r2. (ii) Hence show that (u − v)2 + (v − w)2 + (w − u)2 = 2(q2 − 3pr)/r2.
Using (u − v)2 + (v − w)2 + (w − u)2 = 2(q2 − 3pr)/r2, prove that q²<3pr forces at least one original root α,β,γ to be nonreal.
Consider Fc(x) = x3 + cx2 − 6x + 1, where c is real. You may use the sufficient condition: a real cubic x³+px²+qx+r, r≠0, has a nonreal root if q²<3pr. Show that F₁₃(x)=0 has a nonreal root.
Consider Fc(x) = x3 + cx2 − 6x + 1, where c is real. You may use the sufficient condition: a real cubic x³+px²+qx+r, r≠0, has a nonreal root if q²<3pr. A student claims that q²≥3pr guarantees all roots are real. (i) By varying the positive integer c, find the…