if sin^14A+cos^20A=x then find the smallest interval in which value of x lie

well, since sin^14 and cos^20 both have range [0,1], we know that x must lie in [0,2].

A little graphing shows that in fact x lies in [.006222,1]

Still haven't come up with an analysis to show that max of sin^2m(A) + cos^2n(A) = 1

or exactly what the minimum is.

To find the smallest interval in which the value of x lies, we need to consider the maximum and minimum values that sin^14(A) and cos^20(A) can take.

For sin^14(A), the maximum value is 1 (when A = 90° or π/2) and the minimum value is 0 (when A = 0° or 2π).

Similarly, for cos^20(A), the maximum value is 1 (when A = 0° or 2π) and the minimum value is 0 (when A = 90° or π/2).

To find the smallest interval, we need to find the overlapping range between these two functions. Since both functions are raised to even powers, they can never be negative.

Therefore, the smallest interval in which the value of x lies is [0, 1].