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

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In summary, the graph of y=f(x) is concaved down over the interval (A,B) and 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, which minimizes the area bounded by the curve, the tangent, and lines x=A and x=B, is independent of f(x).
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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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Do you have no clue at all?

Begin with a drawing of the situation. Try to be more specific in your question. We're not going to do the problem for you...
 
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sketch

i have done some sketche
 

1. What is a tangent in math?

A tangent in math is a line that touches a curve at a single point, without crossing or intersecting it. It can also refer to the trigonometric function that represents the ratio of the opposite side to the adjacent side in a right triangle.

2. How do you solve a tangent problem?

To solve a tangent problem, you need to use the tangent function and the given information about the triangle, such as the length of the opposite and adjacent sides. You can then use trigonometric identities and the Pythagorean theorem to find the missing values.

3. What is the difference between a tangent and a secant in math?

A tangent is a line that touches a curve at a single point, while a secant is a line that intersects a curve at two points. In trigonometry, the tangent function represents the slope of the tangent line, while the secant function represents the ratio of the hypotenuse to the adjacent side in a right triangle.

4. What are common real-life applications of tangent in math?

The concept of tangent is used in many real-life applications, such as in architecture to calculate the slope of a roof or the angle of a staircase. It is also used in navigation to determine the distance between two points or the bearing of a ship. Additionally, tangent is used in physics to calculate forces and in engineering to design structures.

5. What are some tips for solving tangent problems?

Some tips for solving tangent problems include drawing accurate diagrams, labeling the given values, using the correct trigonometric function, and double-checking your calculations. It is also helpful to memorize common trigonometric identities and to practice solving different types of tangent problems.

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