Question

In: Math

Given a general cubic function y = ax^3 + bx^2 + cx + d prove the...

Given a general cubic function y = ax^3 + bx^2 + cx + d prove the following,

a) That a cubic must change at a quadratic rate.

         I believe this to be the derivative of the general cubic function, yielding dy/dx = 3ax^2 + 3bx + c

b) That there are only 6 basic forms (shapes) for a cubic.

        This question is where I get lost. Please help, Thanks!

Solutions

Expert Solution

To show that a cubic function must change at a quadratic rate and there are only 6 basic forms of a cubic function.


Related Solutions

1. Consider the cubic function f ( x ) = ax^3 + bx^2 + cx +...
1. Consider the cubic function f ( x ) = ax^3 + bx^2 + cx + d where a ≠ 0. Show that f can have zero, one, or two critical numbers and give an example of each case. 2. Use Rolle's Theorem to prove that if f ′ ( x ) = 0 for all xin an interval ( a , b ), then f is constant on ( a , b ). 3.True or False. The product of...
The curves of the quadratic and cubic functions are f(x)=2x-x^2 and g(x)= ax^3 +bx^2+cx+d. where a,b,c,d...
The curves of the quadratic and cubic functions are f(x)=2x-x^2 and g(x)= ax^3 +bx^2+cx+d. where a,b,c,d ER, intersect at 2 points P and Q. These points are also two points of tangency for the two tangent lines drawn from point A(2,9) upon the parobala. The graph of the cubic function has a y-intercept at (0,-1) and an x intercept at (-4,0). What is the standard equation of the tangent line AP.
The curves of the quadratic and cubic functions are f(x)=2x-x^2 and g(x)= ax^3 +bx^2+cx+d. where a,b,c,d...
The curves of the quadratic and cubic functions are f(x)=2x-x^2 and g(x)= ax^3 +bx^2+cx+d. where a,b,c,d ER, intersect at 2 points P and Q. These points are also two points of tangency for the two tangent lines drawn from point A(2,9) upon the parobala. The graph of the cubic function has a y-intercept at (0,-1) and an x intercept at (-4,0). What is the value of the coefficient "b" in the equation of the given cubic function.
Find the cubic equation: f(x) = ax^3+bx^2+cx+d for which f(-1)=3, f(1)=1, f(2)=6, and f(3)=7. Find the...
Find the cubic equation: f(x) = ax^3+bx^2+cx+d for which f(-1)=3, f(1)=1, f(2)=6, and f(3)=7. Find the value of a, b, c, and d
Find the cubic equation. F(x)=ax^3+bx^2+cx+d F(-1)=3 F(1)=1 F(2)=6 F(3)=7 What is the value of a,b,c,d
Find the cubic equation. F(x)=ax^3+bx^2+cx+d F(-1)=3 F(1)=1 F(2)=6 F(3)=7 What is the value of a,b,c,d
Find a cubic function y = ax3 + bx2 + cx + d whose graph has...
Find a cubic function y = ax3 + bx2 + cx + d whose graph has horizontal tangents at the points (−2, 8) and (2, 2). Find an equation of the normal line to the parabola y = x2 − 8x + 7  that is parallel to the line x − 2y = 2.
Ax = 6m and Ay= -8m, Bx= -8m and By= 3m, Cx = 27m and Cy=...
Ax = 6m and Ay= -8m, Bx= -8m and By= 3m, Cx = 27m and Cy= 21m. Determine a and b such that aA + bB + C = 0. Include a sketch of the scenario.
Show that the map T(a,b,c) = (bx^2 + cx + a) is an isomoprhism between R^3...
Show that the map T(a,b,c) = (bx^2 + cx + a) is an isomoprhism between R^3 and P_2
Use a system of equations to find the parabola of the form y=ax^2+bx+c that goes through...
Use a system of equations to find the parabola of the form y=ax^2+bx+c that goes through the three given points. (2,-12)(-4,-60)(-3,-37)
1252) y=(C1)exp(Ax)+(C2)exp(Bx)+F+Gx is the general solution of the second order linear differential equation:    (y'') +...
1252) y=(C1)exp(Ax)+(C2)exp(Bx)+F+Gx is the general solution of the second order linear differential equation:    (y'') + ( 6y') + (-27y) = ( 2) + ( -3)x. Find A,B,F,G, where A>B. This exercise may show "+ (-#)" which should be enterered into the calculator as "-#", and not "+-#". ans:4
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT