Question
Problem and solution on derivatives
Answers
GPT-4o mini
Certainly! Let's discuss a problem that involves derivatives and how to solve it step by step.
### Problem
Find the maximum and minimum values of the function:
\[ f(x) = 3x^2 - 12x + 7 \]
on the interval \([1, 5]\).
### Solution
To find the maximum and minimum values of the function on the given interval, we will follow these steps:
1. **Find the derivative of the function**:
\[
f'(x) = \frac{d}{dx}(3x^2 - 12x + 7) = 6x - 12
\]
2. **Set the derivative equal to zero to find critical points**:
\[
6x - 12 = 0 \implies 6x = 12 \implies x = 2
\]
3. **Evaluate the critical point and the endpoints of the interval**:
We need to evaluate \(f(x)\) at the critical point \(x = 2\) and the endpoints \(x = 1\) and \(x = 5\).
- For \(x = 1\):
\[
f(1) = 3(1)^2 - 12(1) + 7 = 3 - 12 + 7 = -2
\]
- For \(x = 2\):
\[
f(2) = 3(2)^2 - 12(2) + 7 = 3(4) - 24 + 7 = 12 - 24 + 7 = -5
\]
- For \(x = 5\):
\[
f(5) = 3(5)^2 - 12(5) + 7 = 3(25) - 60 + 7 = 75 - 60 + 7 = 22
\]
4. **Compare the values to find the maximum and minimum**:
- \(f(1) = -2\)
- \(f(2) = -5\)
- \(f(5) = 22\)
- Minimum value: \(-5\) at \(x = 2\)
- Maximum value: \(22\) at \(x = 5\)
### Conclusion
The minimum value of the function \(f(x)=3x^2 - 12x + 7\) on the interval \([1, 5]\) is \(-5\) at \(x = 2\), and the maximum value is \(22\) at \(x = 5\).
### Problem
Find the maximum and minimum values of the function:
\[ f(x) = 3x^2 - 12x + 7 \]
on the interval \([1, 5]\).
### Solution
To find the maximum and minimum values of the function on the given interval, we will follow these steps:
1. **Find the derivative of the function**:
\[
f'(x) = \frac{d}{dx}(3x^2 - 12x + 7) = 6x - 12
\]
2. **Set the derivative equal to zero to find critical points**:
\[
6x - 12 = 0 \implies 6x = 12 \implies x = 2
\]
3. **Evaluate the critical point and the endpoints of the interval**:
We need to evaluate \(f(x)\) at the critical point \(x = 2\) and the endpoints \(x = 1\) and \(x = 5\).
- For \(x = 1\):
\[
f(1) = 3(1)^2 - 12(1) + 7 = 3 - 12 + 7 = -2
\]
- For \(x = 2\):
\[
f(2) = 3(2)^2 - 12(2) + 7 = 3(4) - 24 + 7 = 12 - 24 + 7 = -5
\]
- For \(x = 5\):
\[
f(5) = 3(5)^2 - 12(5) + 7 = 3(25) - 60 + 7 = 75 - 60 + 7 = 22
\]
4. **Compare the values to find the maximum and minimum**:
- \(f(1) = -2\)
- \(f(2) = -5\)
- \(f(5) = 22\)
- Minimum value: \(-5\) at \(x = 2\)
- Maximum value: \(22\) at \(x = 5\)
### Conclusion
The minimum value of the function \(f(x)=3x^2 - 12x + 7\) on the interval \([1, 5]\) is \(-5\) at \(x = 2\), and the maximum value is \(22\) at \(x = 5\).