Answer
A: $\frac{d^{2}}{dx^{2}}(f*g)=((+)(+))+((+)(+))+((+)(+))+((+)(+))=4(+)$, concave up
B: $\frac{d^{2}}{dx^{2}}(f*g)=((+)(+))+((-)(-))+((+)(+))+((-)(-))=4(+)$, concave up
C:
1:$\frac{d^{2}}{dx^{2}}(f*g)=((-)(+))+((-)(+))+((+)(-))+((+)(-))=4(-)$, concave down
2: $\frac{d^{2}}{dx^{2}}(f*g)=((+)(+))+((+)(+))+((+)(-))+((+)(-))=2(+)+2(-)$, concavity depends on first derivatives
3: $\frac{d^{2}}{dx^{2}}(f*g)=((+)(-))+((+)(+))+((+)(-))+((+)(-))= (+)+3(-)$ AND the power of $f*g$ is one, linear, no concavity
Work Step by Step
We first rewrite functions in terms of sums using the product rule for both the first and second derivative.
$\frac{d}{dx}(f*g)=(f*\frac{d}{dx}g)+(\frac{d}{dx}f*g)$
$\frac{d^{2}}{dx^{2}}(f*g)=\frac{d}{dx}((f*\frac{d}{dx}g)+(\frac{d}{dx}f*g))=(f*\frac{d^{2}}{dx^{2}}g)+(\frac{d^{2}}{dx^{2}}f*g)+(\frac{d}{dx}f*\frac{d}{dx}g)+(\frac{d}{dx}f*\frac{d}{dx}g)$
A: If a function is increasing, positive, and concave up, then the function and its first and second derivative must all be positive. Using the formula we used before,
$\frac{d^{2}}{dx^{2}}(f*g)=((+)(+))+((+)(+))+((+)(+))+((+)(+))=4(+)$
and thus will be concave up.
B: If a function is decreasing, positive, and concave up, then the first derivative must be negative, and the main function and second derivative must be positive
$\frac{d^{2}}{dx^{2}}(f*g)=((+)(+))+((-)(-))+((+)(+))+((-)(-))=4(+)$
and thus will be concave up.
C:
1: $f$ is negative concave down and $g$ is positive concave up, $f$ is increasing and $g$ is decreasing, so $\frac{d}{dx}f$ is postitive and $\frac{d}{dx}$ g is negative.
$\frac{d^{2}}{dx^{2}}(f*g)=((-)(+))+((-)(+))+((+)(-))+((+)(-))=4(-)$, and thus will be concave down, depending on the size of the first derivatives.
2:
$f$ is positive concave up and $g$ is positive convave up, $f$ is increasing and $g$ is decreasing, so $\frac{d}{dx}f$ is postitive and $\frac{d}{dx}$ g is negative.
$\frac{d^{2}}{dx^{2}}(f*g)=((+)(+))+((+)(+))+((+)(-))+((+)(-))=2(+)+2(-)$, which means that the function can either be concave up or concave down.
3: $f$ is positive concave up, and $g$ is positive concave down, AND the power of $f*g=1$ $f$ is increasing and $g$ is decreasing, so $\frac{d}{dx}f$ is postitive and $\frac{d}{dx}g$ is negative.
$\frac{d^{2}}{dx^{2}}(f*g)=((+)(-))+((+)(+))+((+)(-))+((+)(-))= (+)+3(-)$, but since the power of the product of $f*g$ is one, it makes the function linear, and thus, have no concavity.
The arguments for A and B do not work for the scenarios in C because the first derivatives in these cases have different signs than in A and B, where they have the same sign.
