Need help with true-false questions and explanations behind them:
Let A be a square matrix. If $det(A^{100}) = 0$, then $det(A) = 0$.
I just dont understand what this question is actually saying. The question is written where it looks like it is $A^{100}$ but I am not sure if it is actually referring to $A$ to the power $100$ and not sure how to go about solving this or proving this if this is true or not. If someone could give me an explanation for this question or provide me with some similar examples and what concepts I need to know, I would greatly appreciate it.
- The function $f:\mathbb{R} \to \mathbb{R}$ defined by $f(x) = | x^2- 1 |$ has one critical point. Note: $|\cdot|$ refers to the absolute value of x
For this one, I am thinking this is false as a function critical points exist when dy/dx = 0 or when it is undefined. This function derivative would be $2x$ and $-2x = 0$ at $x = 0$. But since the function works differently on different points such as when the function is between $-1 < x < 1$ so at $-1$ or $1$ it would be undefined derivative? Therefore there are 3 critical points? I am not sure if that is the right explanation.
- Any tangent to the curve defined by $x^2 + y^2 = 25$ is of the form:
y = -(x0/y0)x + 25/y0,
where $(x_0, y_0)$ is the point of tangency.
I don't understand this question either. I am not sure what this is exactly saying. If x0,y0 are arbitrary points on the curve, so do I find the derivative of the equation and then do what with it? I thought some points there would be no derivative as the curve simply doesn't exist at some points or the derivative would be undefined or the tangent touches the function twice at some points due to the equation defining a circle where a $y-value$ corresponds to $2$ $x-values$ . I am not sure how to prove if this one is correct or not.
- If $g''(a) = 0$, then g must have a point of inflection at a.
This one isn't true because, for point of inflection to exist, $f''(x) = 0$ and the points around it must change in concavity right? so just knowing that the $f''(a) = 0$ doesn't tell you $100%$ that a point of inflection exists there unless you knew that the points before and after, there was a change in concavity i.e from concave down to up or vice versa?