10.        Find the cubic equation f(x) = `+`(`*`(`^`(ax, 3)), `*`(`^`(bx, 2)), cx, d); given there is a maximim at 3, 8;  and a min at -5, -2; .  

 

restart;  

f := proc (x) options operator, arrow, function_assign; `+`(`*`(a, `*`(`^`(x, 3))), `*`(b, `*`(`^`(x, 2))), `*`(c, `*`(x)), d) end proc;  

f := proc (x) options operator, arrow; `+`(`*`(a, `*`(`^`(x, 3))), `*`(b, `*`(`^`(x, 2))), `*`(c, `*`(x)), d) end proc; (6.10.1)
 

f(3);  

`+`(`*`(27, `*`(a)), `*`(9, `*`(b)), `*`(3, `*`(c)), d); (6.10.2)
 

f(-5);  

`+`(`-`(`*`(125, `*`(a))), `*`(25, `*`(b)), `-`(`*`(5, `*`(c))), d); (6.10.3)
 

diff(f(x), x);  

`+`(`*`(3, `*`(a, `*`(`^`(x, 2)))), `*`(2, `*`(b, `*`(x))), c); (6.10.4)
 

 

fp(x); (6.10.5)
 

fp(3);  

`+`(`*`(27, `*`(a)), `*`(6, `*`(b)), c); (6.10.6)
 

fp(-5);  

`+`(`*`(75, `*`(a)), `-`(`*`(10, `*`(b))), c); (6.10.7)
 


 

 

s1 := solve({(`+`(`*`(27, `*`(a)), `*`(6, `*`(b)), c);  

s1 := {a = -`/`(5, 128), b = -`/`(15, 128), c = `/`(225, 128), d = `/`(619, 128)}; (6.10.8)
 

 

assign(s1);  

 

f(x);  

`+`(`-`(`*`(`/`(5, 128), `*`(`^`(x, 3)))), `-`(`*`(`/`(15, 128), `*`(`^`(x, 2)))), `*`(`/`(225, 128), `*`(x)), `/`(619, 128)); (6.10.9)
 

 

f(x); (6.10.10)
 

 

f(3);  

8; (6.10.11)
 

f(-5);  

-2; (6.10.12)
 

with(Student[Calculus1]); -1; ExtremePoints(f(x)) 

[-5, 3]; (6.10.13)