Maximum and minimum values, power functions
Find the greatest and least value of a function on a set, compare the graphs of y = xⁿ for even and odd n, and meet the graph of y = x⁻ⁿ.
- أوكرانيا: الصف ١٠
- روسيا: الصف ١٠
- ألمانيا: الصف ٩
- فرنسا: الصف ١٠
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الدليل
Ukrainian Grade 10 (algebra and calculus, §1, sections 1 to 3): greatest and least values, power functions with a natural exponent and with an integer exponent.
f(x₀) is the greatest value of f on a set M if x₀ ∈ M and f(x) ≤ f(x₀) for every x ∈ M. It is written maxf(x) = f(x₀), x ∈ M; the least value is min. A function may have no greatest or no least value on a set.
For y = xⁿ, n a natural number: for even n the function is even, decreasing on ]−∞، ٠] and increasing on [٠، +∞[; for odd n it is odd and increasing on ℝ.
y = x⁻ⁿ is defined for x ≠ 0. For even n it is even with range ]٠، +∞[; for odd n it is odd with range ]−∞، ٠[ ∪ ]٠، +∞[.