Concave-up and concave-down are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

#4
What happens to concavity when functions are added?
a) If f(x) and g(x) are concave up for all x, is f(x) + g(x) concave up for all x?
Yes
b) If f(x) is concave up for all x and g(x) is concave down for all x, what can you say about the concavity of f(x) + g(x)? For example, what happens if f(x) and g(x) are both polyn

1. Draw a sign graph to determine where the following function is increasing or decreasing. Identify all stationary points.
y=2x^4-4x^2+1
2. Find the intervals where the following curve is concave up or down. Also find the coordinates of any points of inflection.
y=2x^3 - x^2 +3x+5
3

1.) Find and equation of the tangent to the curve at the point corresponding to the given value of the parameter
x= cos t + sin 2t, y= sin t + cos 2t (t=0)
2.) Find dy/dx and d^2/dx^2 for which values of t is the curve concave upward
x= t + ln t, y = 1 - ln t

Let f(x)=x^(4)รข?'6x^(3)+12x^(2). Find (a) the intervals on which f is increasing, (b) the intervals on which f is decreasing, (c) the open intervals on which f is concave up, (d) the open intervals on which f is concave down, and (e) the x-coordinates of all inflection points.
(a) f is increasing on the interval(s) =
(b)

This problem deals with lenses, please explain using equations, and ray diagrams.
A real object is at the zero end of the meter stick. A large concave mirror at the 100-cm end of a meter stick forms an image of the object at the 70-cm position. A small convex mirror placed at the 20-cm position forms a final image at the 10-c

Which kind of spherical mirror, concave or convex, can be used to start a fire with sunlight? For the best results, how far from the mirror should the paper to be ignited be placed?

I am taking a course by distance, and my professor provided an example of how to create a Hessian matrix using partial derivatives. He gave another example that just had the solution for us to try on our own.
I think that I am somehow not taking the second order partial derivative right. The attached file has the professor