Minimizing Area: Proof that k is Independent of f(x) in a Tangent Problem

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SUMMARY

The discussion centers on the mathematical proof that the value of k, which minimizes the area bounded by a concave down curve y=f(x) and its tangent line at point P(k, f(k)), is independent of the specific function f(x) over the interval [A, B]. Participants emphasize the importance of understanding the conditions f''(x) < 0 and the geometric interpretation involving points P and Q where the tangent intersects the lines x=A and x=B. The conclusion drawn is that the minimization of the area is a property of the tangent line's position rather than the function itself.

PREREQUISITES
  • Understanding of concave functions and their second derivatives, specifically f''(x) < 0.
  • Familiarity with tangent lines and their geometric properties in calculus.
  • Basic knowledge of area calculations under curves in integral calculus.
  • Ability to sketch and interpret graphs of functions and their tangents.
NEXT STEPS
  • Study the properties of concave functions and their implications in optimization problems.
  • Learn about the geometric interpretation of derivatives and tangents in calculus.
  • Explore integral calculus techniques for calculating areas under curves.
  • Investigate related optimization problems in calculus, focusing on independent variables.
USEFUL FOR

Mathematics students, calculus instructors, and anyone interested in optimization problems involving concave functions and tangents.

elsen_678
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the graph of y=f(x) is concaved down over the interval (A,B)
ie f''(x)<0 for A=<x=<B
the tangent of the curve at point P(k,f(k)) meets lines x=A and x=B at P and Q respectively.
the value of k is such that the area bounded by the curve, the tangent and lines x=A and x=B is minimized.
Prove that k is independent of f(x).

somebody helps thanks
 
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