?Yes, you are right. The function has 3 roots. I compute, that other roots is x2-epsilon, and x3+epsilon,
where
epsilon~~10^-3559
best
Aleksas D?
Here is my calculation of x3+epsilon
(%i1) display2d:false;
(out1) false
(%i2) eq1:log(x^3-x*78+90)+1000*x;
(out2) log(x^3-78*x+90)+1000*x
(%i3) eq2:exp(%);
(out3) (x^3-78*x+90)*%e^(1000*x)
(%i4) eq3:x^3-78*x+90;
(out4) x^3-78*x+90
(%i5) rootsepsilon:10^-15;
(out5) 1/1000000000000000
(%i6) float(realroots(eq3));
(out6) [x = -9.360292928323476,x = 1.174624074847539,x = 8.185668853475937]
(%i7) fpprec:10000;
(out7) 10000
(%i8) q:bf_find_root(eq3,x,8.9,8.0,relerr=10^-9000)$
(%i9) /* p:bf_find_root(eq2,x,8.9,8.0,relerr=10^-9000)$ takes too long, still waiting on this one */ a:%e^(-1000*x),x=q$
(%i10) eq5:df=diff(eq3);
(out10) df = (3*x^2-78)*del(x)
(%i11) eq6:solve(eq5,del(x));
(out11) [del(x) = df/(3*x^2-78)]
(%i12) (block[fpprec:fpprec], p:at(eq6,[df=a,x=q]), fpprec:50, bfloat(p));
(out12) [del(x) = 8.3028334845888254974744577740370102072586144034212b-3558]
I am curious to see if this is exact, to 50 digits. I have to wait for the calculation of p.
Rich