Let K be the closed interval [0,1] and consider the function f(x)=x^2. Is f convex? Is f linear?
help please :/ i don't even know how to set this up to check, our teacher didn't even get to this in class yet!
help please :/ i don't even know how to set this up to check, our teacher didn't even get to this in class yet!
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f(x) = x²
"Is f linear?": No. The term "linear" means the same as "first order" or "first degree" meaning that no term in the expression has an order greater than 1. In this expression the x² term has order 2 so the function is quadratic, or 2nd order, not linear or linear.
Is f convex? Yes or No, depending on viewing direction. Since the slope of the function's graph = 2x, the slope increases monotonically over the interval meaning the graph curves upward as you go from x = 0 to x = 1. Is that convex? If you are looking from the bottom, you see a convex curve, curving away from you. If you are looking from the top, it is concave, curving toward you. It is less ambiguous to say that the curve is concave upward.
"Is f linear?": No. The term "linear" means the same as "first order" or "first degree" meaning that no term in the expression has an order greater than 1. In this expression the x² term has order 2 so the function is quadratic, or 2nd order, not linear or linear.
Is f convex? Yes or No, depending on viewing direction. Since the slope of the function's graph = 2x, the slope increases monotonically over the interval meaning the graph curves upward as you go from x = 0 to x = 1. Is that convex? If you are looking from the bottom, you see a convex curve, curving away from you. If you are looking from the top, it is concave, curving toward you. It is less ambiguous to say that the curve is concave upward.
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The slope is always 2*x, so it is not linear, yes the function is convex, meaning the slope increases as x increases.