Consider y = f(x) a real value function for all x€ R. It is given that graph of the function f(x) is symmetric about the lines x = a and x = b, (b > a) .
Prove that f(x) is periodic.
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Answers
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Answer:
Test statement #1.
f(x) = -0.3(x - 5)² + 5
This is the equation of a parabola with vertex at (5,5).
Therefore the function is symmetric about x=5.
The statement "The axis of symmetry is x=5" is TRUE.
Test statement #2.
f(x) is defined for all real values of x.
The statement "The domain is {x | x is a real nuber} is TRUE.
Test statement #3.
As x -> -∞, f(x) -> -∞.
f(5) = -0.3*(5-5)^2 + 5 = 5
Therefore f(x) is creasing over (-∞, 5) is TRUE.
Test statement #4.
As x -> +∞, f(x) -> -∞.
Therefore the curve is concave downward., and it has no minimum.
The statement "The minimum is (5,5)" is False.
Test statement #5.
The maximum value of f(x) occurs at the vertex because the curve is concave downward.
The statement "The range is {y | y≥5}" is False.