IIT JEE Function Maxima  Summary and Q&A
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.
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 realvalued 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.