Find the equation of the inflexion tangent to a curve y=(30*x^3 + 40)* e^3?

Question

correction: y=(30*x^3 + 40) * e^(3*x)

Answer 1

Take the first and second derivatives. Set the second derivative equal to zero and solve to find the point of inflection.

y' = 90x^2 e^(3x) + 3(30x^3 + 40) e^(3x) = e^(3x) (90x^3 + 90x^2 + 120)

y'' = 3e^(3x) (90x^3 + 90x^2 + 120) + e^(3x) (270x^2 + 180x) = e^(3x) (270x^3 + 540x^2 + 180x + 360) = 0

e^(3x) ≠ 0 for all real numbers.

270x^3 + 540x^2 + 180x + 360 = 0

3x^3 + 6x^2 + 2x + 4 = 0
3x^2(x + 2) + 2(x + 2) = 0
(3x^2 + 2)(x + 2) = 0
So x = -2.

The tangent line slope at x = -2 is y'(-2) = -0.595

The tangent line passes through the point of inflection (-2, y[-2]) = (-2, -0.496)

This means the tangent line is y + 0.496 = -0.595(x + 2).

Answer 2

A point of inflection has the properties that its first 2 derivatives vanish.

y(x) = (30*x^3 + 40)* e^3
y'(x) = e^3*90x^2
y''(x) = e^3*180x

y'(x) = y''(x) = 0 when x=0

y(0) = 40e^3

Normally the equation of a line is y=mx+b where m is the slope and b is the y-intercept. Since m=0 and b=40e^3 the equation of the line is

y=40e^3

Answer 3

y´=e^3*(90x^2) and y´´= e^3*180x =0 so x=0
sign of y´´ --------0++++ so x=0 and y= 40e^3 is an inflexion point
y´(0) =0 so the tangent is y= 40e^3
now I see your correction
y´=e^3x(90x^2)+3e^3x(30x^3+40)=e^3x(90x^3+90x^2+120)
=30e^3x(3x^3+3x^2+4)
y´´=30[e^3x(9x^2+6x)+3e^3x(3x^3+3x^2+4)]=
=30e^3x(9x^3+18x^2+6x+12)
3x^3+6x^2+2x+4=0 which canbe factore as(x+2)(3x^2+2)=0
so the only real root is x=-2
The inflexion point is (-2,-200e^-8)
y´(-2)=30*e^-8 (-8)= -240e^-8
The tangent is y+200*e^-8 = -240e^-8(x+2)

Answer 4

The inflection point(s) can be found by differentiating the function

In this case, you get: 3e^3x*(30x^3+40) + e^3x*(90x^2) =

e^3x * (90x^3 + 90x^2 + 120) = 30e^3x*(3x^3+3x^2 + 4)

Now, set this equal to 0, and that's where the inflection occurs. So, you get 30e^3x*(3x^3+3x^2+4) = 0 or

e^3x*(3x^3+3x^2+4) = 0

Answer 5

im thinkin it might be when the derivative of that equals 0 but dont hold me to it