How do you find the area of the largest isosceles triangle having a perimeter of 18 meters?

How do Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem relate to the Fundamental Theorem of Calculus for ordi-nary single integral?

Use a Numerical approach (i,e., a table of values) to approximate ${\displaystyle \underset{x\to 0}{lim}\frac{e^{2}x-1}{x},}$ Provide support for your analysis (slow the table and explain your resoning).

${\displaystyle {\int }_1^0\tan \left(x^2\right)dx+2{\int }_1^0{x}^2{\sec }^2\left(x^2\right)dx=\frac{\pi }{4}}$
Select one:
True
False

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?

Let ${\displaystyle f\left(x\right)=x^3-6x^2-15x+23}$
a) Perform a full sign analysis on ${\displaystyle f\left(x\right):}$ find its zeroes, and determine its sign on each interval.
b) Use the information in (2) to determine intervals of concave up / concave down for f.
c) Find the coordinates (x, y) of the inflection point(s) of f.
d) Find the absolute maximum and minimum values of f on the interval ${\displaystyle [0,7]}$. Justify your answer with appropriate calculus work.

Define variables, write systems of equations, write coefficient matrix, the B matrix and the solution matrix, x
${\displaystyle A=[],B=[],X=[]}$

Exercise No. 10 (D.E. with coefficient linear in two variables)
Find the general / solution of the following D.E.
4. $(2x+3y-5)dx+(3x-y-2)dy=0$