A sodium chloride crystal in the shape of a cube is expanding at the rate of 60 cubic microns per second. How fast is the side of the cube growing when the volume is 1000 cubic microns?

Second derivatives For the following sets of variables, find all the relevant second derivatives. In all cases, first find general expressions for the second derivatives and then substitute variables at the last step.
${\displaystyle f(x,y)=x^{2}y-x{y}^2\text{ ,where }x=st\text{ and }y=\frac{s}{t}}$

multivariable functions works when the partial derivatives are continous but if the function is just differentiable does the chain rule work ? I mean if ${\displaystyle z=f(x,y)}$ is differentiable function and ${\displaystyle x=g\left(t\right),y=h\left(t\right)}$ are also differentiable functions can we write:
${\displaystyle \frac{dz}{dt}=\frac{\partial f}{\partial x}\times \frac{dx}{dt}+\frac{\partial f}{\partial y}\times \frac{dy}{dt}}$

A camera shop stocks eight different types of batteries, one of which is type A7b.Assume there are at least 30 batteries.
of each type.a. How many ways can a total inventory of 30 batteries be distributed among the eight different types? b.
How many way can a total inventory of 30 batteries be distributed among the eight different types of the inventory must
include at least four A76 batteries?c. How many ways can a total inventory of 30 batteries be distributed among the eight
different types of the inventory includes at most three A7b batteries

This might be trivial but I am struggling to justify the following simplification.from:
$h(x^{\ast }+d)-h(x^{\ast })\ge \sum _{j\text{⧸}\in G}{\mathrm{\nabla }}_{j}f(x^{\ast })\ast d_j+\gamma \sum _{j\text{⧸}\in G}|d_j|$
to:
$h(x^{\ast }+d)-h(x^{\ast })\ge -\underset{j\text{⧸}\in G}{max}|{\mathrm{\nabla }}_{j}f(x^{\ast })|\sum _{j\text{⧸}\in G}|d_j|+\gamma \sum _{j\text{⧸}\in G}|d_j|$

Specifically, why is there a negative in front of the maximization?

Note:
I can get behind the fact that
$\sum _{j\text{⧸}\in G}{\mathrm{\nabla }}_{j}f(x^{\ast })\ast d_j\ge \underset{j\text{⧸}\in G}{max}|{\mathrm{\nabla }}_{j}f(x^{\ast })|\sum _{j\text{⧸}\in G}|d_j|$
provided the jacobian is nonnegative element-wise. But then why add the negative sign?

Multivariable optimization question. Find three positive real numbers whose sum is one and the sum of their squares is a minimum.

Answer the following questions about an ANOVA analysis involving three samples.
a. In this ANOVA analysis, what are we trying to determine about the three populations they’re taken from?
b. State the null and alternate hypotheses for a three-sample ANOVA analysis.
c. What sample statistics must be known to conduct an ANOVA analysis?
d. In an ANOVA test, what does an F test statistic lower than its critical value tell us about the three populations we’re examining?