What is the minimum value of $f(x)$ with positive real numbers $p,q,r$?

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In summary, the minimum value of a function $f(x)$ is the lowest output that the function can produce for any input value, represented by $f_{min}(x)$ or $min\{f(x)\}$. To find the minimum, various methods such as differentiation, graphing, or algebraic manipulation can be used. The minimum is significant in applications like optimization problems. A function $f(x)$ can only have one minimum value, and the presence of parameters $p,q,r$ can affect its position or value.
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anemone
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For positive real numbers $p,\,q,\,r$, determine the minimum of the function $f(x)=\sqrt{p^2+x^2}+\sqrt{(q-x)^2+r^2}$.
 
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anemone said:
For positive real numbers $p,\,q,\,r$, determine the minimum of the function $f(x)=\sqrt{p^2+x^2}+\sqrt{(q-x)^2+r^2}$.

Hint:

The proposed solution uses the geometry approach that solved it neatly and nicely.:)

Further hint:

Minimum $f(x)$ is $\sqrt{(p+r)^2+q^2}$.
 
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Solution of other:

View attachment 4563

Let $AD=p,\,AE=q,\,EC=r$. Let $B$ be a point on $AE$ and let $x=AB$, so that $BE=q-x$. Then $f(x)=DB+BC$. To minimize $DB+BC$, we use the method of reflection. Let $C'$ be the reflection of $C$ in the line $AE$.

Since triangles $BEC$ and $BEC'$ are congruent, $BC=BC'$ and $f(x)=DB+BC'$.

As $x$ varies, $B$ changes its position. But the distance $DB+BC'$ will be a minimum when $B$ lies on the line $DC'$ (as shown in the orange line).

The minimum value of $f(x)$ is then $DB+BC'=DC'$. Let $DD'$ be the perpendicular from $D$ to the line $CC'$. From the right triangle DD'C',

$f(x)_{\text{minimum}}=DC'=\sqrt{DD'^2+D'C'^2}=\sqrt{q^2+(p+r)^2}$
 

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FAQ: What is the minimum value of $f(x)$ with positive real numbers $p,q,r$?

1. What is the minimum value of the function $f(x)$?

The minimum value of a function $f(x)$ is the lowest output that the function can produce for any input value. It is represented by the notation $f_{min}(x)$ or $min\{f(x)\}$.

2. How do you find the minimum of a function $f(x)$?

To find the minimum of a function $f(x)$, you can use various methods such as differentiation, graphing, or algebraic manipulation. Differentiation involves finding the derivative of the function and setting it equal to 0 to solve for the input value that gives the minimum output. Graphing involves plotting the function on a graph and visually identifying the lowest point on the graph. Algebraic manipulation involves manipulating the function to find the input value that gives the minimum output.

3. What is the significance of the minimum of a function $f(x)$?

The minimum of a function $f(x)$ is significant because it represents the lowest possible value that the function can produce, which can be important in various applications. For example, in optimization problems, finding the minimum of a cost function can help determine the most cost-effective solution.

4. Can a function $f(x)$ have multiple minimum values?

No, a function $f(x)$ can only have one minimum value. This is because the minimum value represents the lowest possible output for any input value, and by definition, there can only be one lowest value.

5. How does the presence of parameters $p,q,r$ affect the minimum of a function $f(x)$?

The presence of parameters $p,q,r$ in a function $f(x)$ can affect the minimum by shifting the position of the minimum or changing its value. For example, changing the value of $p$ in a quadratic function can shift the position of the minimum on the x-axis. Similarly, changing the values of $q$ and $r$ in a cubic function can change the minimum value of the function.

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