The graph of the function y= x3+6x2+7x-2cos x changes concavity at x=
x= the moment when someone realizes they're sitting on a whoopee cushion
To determine the point at which the graph of the function changes concavity, we need to find the inflection point. An inflection point is a point on the graph where the concavity changes from concave up (opening upwards) to concave down (opening downwards), or vice versa.
To find the inflection point, we need to find the second derivative of the function and set it equal to zero.
First, let's find the first derivative:
y = x^3 + 6x^2 + 7x - 2cos(x)
dy/dx = 3x^2 + 12x + 7 + 2sin(x)
Next, let's find the second derivative:
d^2y/dx^2 = 6x + 12 + 2cos(x)
Now, let's set the second derivative equal to zero and solve for x:
6x + 12 + 2cos(x) = 0
Unfortunately, it is not possible to find an exact solution for x algebraically. However, we can approximate the solution using numerical methods or software.
To determine where the graph of a function changes concavity, we need to find the points where the second derivative of the function changes sign. In this case, we have to find where the second derivative of the function y = x^3 + 6x^2 + 7x - 2cos(x) changes sign.
Let's find the first derivative of the function:
y' = 3x^2 + 12x + 7 + 2sin(x)
Now, let's find the second derivative by differentiating the first derivative:
y'' = 6x + 12 + 2cos(x)
To find the points where the concavity changes, we need to find the values of x for which y'' equals zero or does not exist.
Setting y'' = 0 and solving for x:
6x + 12 + 2cos(x) = 0
This equation cannot be solved algebraically, so we need to use numerical methods or graphing software to approximate the solution. By using a graphing calculator or software, we can find that the point where the concavity changes is approximately x ≈ -1.625.
Therefore, the graph of the function y = x^3 + 6x^2 + 7x - 2cos(x) changes concavity at approximately x = -1.625.
Concavity changes sign where the second derivative is zero. That would be where
6x + 12 + 2 cosx = 0, which can also be written
3x + 6 + cosx = 0
This transcendental equation will have to be solved numerically by iteration.
If x = -2, 3x + 6 + cosx = -0.42
If x = -1.9, 3x + 6 + cosx = -0.02
-1.9 is close to the correct answer