In the diagram below, PS is a tangent to circle O at point S, POR is a secant, PS =x, PQ = 3, and PR =x+18. X. R. S. (Not drawn to scale) What is the length of PS?
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![**Extra Credit**
In the diagram below, \( \overline{PS} \) is a tangent to circle \( O \) at point \( S \). \( \overline{PQR} \) is a secant. \( PS = x \), \( PQ = 3 \), and \( PR = x + 18 \).
\[
\begin{align*}
&\hspace{5mm} \quad P \\
&\hspace{15mm}/ \quad \backslash \\
&\hspace{10mm} / \quad \quad \backslash \\
&\hspace{5mm} / \quad \quad \quad \backslash \\
&O \quad \quad \quad \quad \backslash \\
& \overline{\ ENS} \quad \quad \\
& \overline{\ ENS} \quad \quad \\
&R \quad (Not\ drawn\ to\ scale) \\
&S \\
\end{align*}
\]
*What is the length of \( \overline{PS} \)?*
**Explanation of the diagram:**
- The circle is labeled \( O \) and has a point \( P \) outside the circle.
- \( S \) is the point on the circle where the tangent \( \overline{PS} \) touches.
- A secant line, \( \overline{PQR} \), intersects the circle at points \( Q \) and \( R \).
- The lengths given are:
- \( PS = x \)
- \( PQ = 3 \)
- \( PR = x + 18 \)
**Objective:**
To find the length of \( \overline{PS} \).
**Important Concepts**:
1. The Tangent-Secant Theorem: For a circle, if a tangent from an external point and a secant from the same external point is drawn, then the square of the length of the tangent is equal to the product of the lengths of the entire secant and its external segment.
\[ \overline{PS}^2 = \overline{PQ} \cdot \overline{PR} \]
2. Using the given values:
\[ x^2 = 3 \cdot (x + 18) \]
Solve for \( x \):
\[
x^2 = 3x + 54 \\
x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe489ebd6-5c8e-44bd-8a5f-33d2b46a9596%2F60c8c773-acdf-43b5-b59b-33187272f5db%2Fg0ju86s_processed.jpeg&w=3840&q=75)
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