- #1
calorimetry
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Homework Statement
Consider a function of one variable, f(x), which is continuous on an interval I. If f has a single critical point, x = a, in I, then if that point is a local maximum (or minimum) it must also be an absolute maximum. Explain why this is.
Homework Equations
Intermediate Value Theorem
Rolle's Theorem
The Attempt at a Solution
Critical point x=a, in I, is a local maximum or minimum, thus f'(x)=0 at x=a.
The function f thus must either concave up (for a to be a local min) or concave down (for a to be a local max).
Since both cases are similar, let consider x=a is a local max.
For a local max, f'(x) is decreasing to 0 as x approaches a from the left and f'(x) is increasing negative away from 0 as x deviate away from a on the right.
Since there is only one single critical point at x=a, the concavity of the graph will not change. f(x) on the left of is an increasing function to f(a) and f(x) on the right of f is a decreasing function away from f(a). Thus x=a must also be the absolute maximum.
Is my reasoning correct? Or did I unconsciously assumed something that I shouldn't have?