IIT JEE Function Maxima

TL;DR

The maximum values of three functions, f, g, and h, on an interval are compared, and it is determined that they are all equal.

Transcript

"Let f, g, and h be real-valued functions defined on the interval, 0 to 1." And it includes the boundaries 0 and 1. That's what the brackets tell us. It wouldn't include the boundaries if we have parentheses. And then they define the functions. f of x is equal to e to the x squared plus e to the negative x squared. g of x looks very similar to f of... Read More

Key Insights

• 😀 The functions f, g, and h satisfy h(x) <= g(x) <= f(x) over the interval 0 to 1.
• 😀 The maximum values of f, g, and h are all equal.
• ☺️ The maximum value of f, g, and h occurs at x = 1.
• 🖐️ The coefficients of the functions play a role in determining their maximum values.
• ☺️ The slope of f(x) is always positive over the interval, leading to its maximum value at x = 1.
• 😀 The maximum values of g and h are equal to the maximum value of f due to their similar expressions and coefficients.
• 😀 The maximum values of f, g, and h cannot be exceeded by any other values over the interval.

Questions & Answers

Q: How do the functions f, g, and h relate to each other over the interval?

Over the interval 0 to 1, the functions satisfy h(x) <= g(x) <= f(x). This is due to the differences in their coefficients.

Q: What are the maximum values of the functions f, g, and h?

The maximum value of all three functions f, g, and h is equal and occurs at x = 1. This can be verified by evaluating the functions at the boundaries of the interval.

Q: How is the maximum value of f determined?

The maximum value of f is determined by evaluating the function at the interval boundaries and comparing the values. The function is found to be increasing over the interval, leading to its maximum value at x = 1.

Q: Why are the maximum values of g and h also equal to the maximum value of f?

The maximum values of g and h are equal to the maximum value of f because their expressions and coefficients are similar to that of f. This means they can both attain the same maximum value, which occurs at x = 1.

Summary & Key Takeaways

• Three real-valued functions, f, g, and h, are defined on the interval 0 to 1.

• The functions are compared based on their coefficients and it is determined that h(x) <= g(x) <= f(x) over the interval.

• The maximum values of f, g, and h are found to be equal and occur at x = 1.