Untitled

Given the function $f(x) = egin{cases} ax+1 & ext{if } x \le 3 \ bx+3 & ext{if } x > 3 \end{cases}$, which of the following relationships between $a$ and $b$ ensures that $f$ is continuous at $x = 3$?

Consider the piecewise function: $f(x) = egin{cases} kx^2 & ext{if } x \le 2 \ 3 & ext{if } x > 2 \end{cases}$. What value of $k$ will make the function continuous at $x=2$?

Determine the value of $k$ that makes the following function continuous at $x = \pi$: $f(x) = egin{cases} kx+1 & ext{if } x \le \pi \ \cos x & ext{if } x > \pi \end{cases}$

For what value of $\lambda$ is the function $f(x) = egin{cases} [\lambda^2x - 2x] & ext{if } x \le 0 \ 4x+1 & ext{if } x > 0 \end{cases}$ continuous at $x=0$?

If $f(x) = egin{cases} rac{k \cos x}{\pi - 2x} & ext{if } x < rac{\pi}{2} \ 3 & ext{if } x = rac{\pi}{2} \end{cases}$ is continuous at $x = rac{\pi}{2}$, find the value of $k$.

Consider the function $f(x) = egin{cases} 5 & ext{if } x \le 2 \ ax+b & ext{if } 2 < x < 10 \ 21 & ext{if } x \ge 10 \end{cases}$. What are the values of $a$ and $b$ that make $f(x)$ a continuous function?

Determine the points of discontinuity for the function $f(x) = x - [x]$, where $[x]$ represents the greatest integer function.

Given $(\cos x)^y = (\cos y)^x$, find $\frac{dy}{dx}$.

Consider the definite integral $\int_{a}^{b} f(x) dx$. If this integral is split at a point 'c' within the interval [a, b], which of the following statements regarding the relationship between the original integral and the resulting integrals is most accurate?

The property $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx$ is applied to simplify integrals. Suppose you are evaluating $\int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \sqrt{\tan x}} dx$. What is the most effective next step after applying this property?

When using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$, for what type of functions $f(x)$ does this transformation provide the most significant advantage in simplifying the integral?

Given that $\int_{0}^{2a} f(x) dx = 2\int_{0}^{a} f(x) dx$ if $f(2a-x) = f(x)$ and $\int_{0}^{2a} f(x) dx = 0$ if $f(2a-x) = -f(x)$, how does the symmetry of $f(x)$ around $x=a$ affect the integral's property?

If $f(x)$ is an even function, $\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx$. If $f(x)$ is odd, $\int_{-a}^{a} f(x) dx = 0$. How can you leverage these properties to evaluate $\int_{-2}^{2} (x^3 + \cos x + 1) dx$ efficiently?

In solving a Linear Programming Problem (LPP), what is the fundamental role of the constraints in determining the optimal solution?

When solving a minimization LPP using the graphical method, how does the objective function's gradient relate to finding the optimal solution at a corner point of the feasible region?

Consider an LPP where the feasible region is unbounded. What implications does this unboundedness have on the solution of the LPP?

Given the constraints $x + 2y \le 12$, $2x + y \le 12$, $4x + 5y \ge 20$, $x \ge 0$, and $y \ge 0$, how does altering the first constraint to $x + 2y \le 10$ affect the feasible region and the optimal solution for maximizing $Z = 600x + 400y$?

Consider the linear programming problem: Minimize $Z = 3x + 5y$ subject to $x + 3y \ge 3$, $x + y \ge 2$, $x \ge 0$, $y \ge 0$. How would incorporating an additional constraint, $x \le 1$, influence the minimum value of $Z$?

Given $A = \begin{bmatrix} 5 & 6 \ 4 & 3 \end{bmatrix}$ and it satisfies $A^2 - 8A - 9I = 0$, which of the following methods is most efficient for computing $A^{-1}$?

If matrix $A = \begin{bmatrix} 2 & -1 & 1 \ -1 & 2 & -1 \ 1 & -1 & 2 \end{bmatrix}$ satisfies $A^3 - 6A^2 + 9A - 4I = 0$, what is the most efficient way to calculate $A^{-1}$?

If matrix $A = \begin{bmatrix} 1 & 3 & 3 \ 1 & 4 & 3 \ 1 & 3 & 4 \end{bmatrix}$, and given that $A \ adj A = |A|I$, how does this relationship simplify the computation of $A^{-1}$?

Suppose $A$ and $B$ are two $2 \times 2$ matrices such that $(AB)^{-1}$ exists. If you find that $B^{-1}A^{-1}$ does not equal $(AB)^{-1}$ after computation, what can you conclude?

Given the function $f(x) = \begin{cases} kx+1 & \text{if } x \le 5 \ 3x-5 & \text{if } x > 5 \end{cases}$, what condition must $k$ satisfy for $f(x)$ to be continuous at $x = 5$, and how does this condition relate to the differentiability of $f(x)$ at $x = 5$?

Consider a function $f(x)$ defined piecewise. What specific criteria must be met at a point $x = c$ within its domain for $f(x)$ to be differentiable at that point?