Maximizing function at value

TL;DR

Find the maximum value of the function f(x) at x=10, given that f(-2) = 3 and f'(x) ≤ 7 for all x.

Transcript

Let f be a differentiable function for all x. If f of negative 2 is equal to 3 and f prime of x is less than or equal to 7 for all x, then what is the largest possible value of f of 10? And so I encourage you to think about this on your own, pause the video, try to figure out the largest possible value for f of 10. And then we'll work through it to... Read More

Key Insights

• ☠️ The problem involves maximizing a function while considering the constraints of a given value and a limit on the instantaneous rate of change.
• 👻 Assuming a constant slope of 7 allows for the determination of the largest possible value of f(10).
• 😫 The condition f'(x) ≤ 7 sets an upper limit on the rate at which the function can grow.

Questions & Answers

Q: How can the problem of finding the maximum value of f(10) be approached?

To maximize f(10), we look for the largest possible slope, which is 7, and assume it remains constant. This allows us to calculate the change in y over the change in x to determine the value of f(10).

Q: Why is it important that the instantaneous rate of change is always less than or equal to 7?

The condition f'(x) ≤ 7 ensures that the slope of the graph of f(x) does not exceed 7 at any point. This sets an upper limit on the rate at which f(x) can increase.

Q: How does assuming a line with a slope of 7 help determine the maximum value of f(10)?

By assuming a slope of 7, we maximize the change in y as x increases. We calculate the change in y over the change in x and find that f(10) - 3 = 84. Solving for f(10), we get f(10) = 87.

Q: What would happen if the slope were lower than 7?

A lower slope would result in a smaller change in y over the change in x. This would lead to a lower value of f(10), as the function would not be increasing as rapidly.

Summary & Key Takeaways

• The problem involves finding the largest possible value of f(10) while satisfying certain constraints.

• The graph of y=f(x) is drawn with the point (-2, 3) and the condition that f'(x) ≤ 7 for all x.

• By assuming the fastest growing function with a slope equal to 7, the maximum value of f(10) is determined to be 87.