2 multiple choice 1 point find the locations ...
2 multiple choice 1 point find the locations of all local maxima and local minima of the function f(x)=4x - 2e^x. local minimum at x = ln(2) local maximum at x = ln(2) and local minimum at x=-ln(2) local maximum at x = 5 local minimum at x = ln(5) local maximum at x = ln(2) clear my selection 3 multiple choice 1 point which of the following statements about functions is true? all functions have a least one local minimum all critical points of a function f(x) give either a local maximum or a local minimum all functions have at least one local maximum critical points are found by finding the values of x which make f(x) equal to zero or undefined. clear my selection
Answer
### 2.
# Explanation:
## Step1: Find the first - derivative
The derivative of $f(x)=4x - 2e^{x}$ is $f^\prime(x)=4 - 2e^{x}$ using the rules $\frac{d}{dx}(ax)=a$ and $\frac{d}{dx}(e^{x})=e^{x}$.
## Step2: Set the first - derivative equal to zero
Set $f^\prime(x)=0$, so $4 - 2e^{x}=0$. Then $2e^{x}=4$, and $e^{x}=2$. Taking the natural logarithm of both sides, we get $x = \ln(2)$.
## Step3: Find the second - derivative
The second - derivative $f^{\prime\prime}(x)=\frac{d}{dx}(4 - 2e^{x})=- 2e^{x}$.
## Step4: Evaluate the second - derivative at the critical point
Substitute $x = \ln(2)$ into $f^{\prime\prime}(x)$. $f^{\prime\prime}(\ln(2))=-2e^{\ln(2)}=-4<0$. Since $f^{\prime\prime}(\ln(2))<0$, the function has a local maximum at $x=\ln(2)$.
# Answer:
Local maximum at $x = \ln(2)$
### 3.
# Explanation:
- Not all functions have a local minimum or local maximum. For example, $y = x$ has no local extrema.
- Critical points of a function $y = f(x)$ are found by setting $f^\prime(x)=0$ or where $f^\prime(x)$ is undefined. A critical point is not necessarily a local maximum or minimum (e.g., $y = x^{3}$ has a critical point at $x = 0$ which is an inflection point).
# Answer:
Critical points are found by finding the values of $x$ which make $f^\prime(x)$ equal to zero or undefined.