Question

The minimum value of \((\mathrm{x}+\mathrm{a})^{2}+(\mathrm{x}+\mathrm{b})^{2}+(\mathrm{x}+\mathrm{c})^{2}\) will be at \(\mathrm{x}\) equal to (a) \([(a+b+c) / 3]\) (b) \([\\{-(a+b+c)\\} / 3]\) (c) \(\sqrt{(a b c)}\) (d) \(a^{2}+b^{2}+c^{2}\)

Short answer

The short answer based on the step-by-step solution is: The minimum value of \((x+a)^2 + (x+b)^2 + (x+c)^2\) will be at \(x = -\frac{a + b + c}{3}\), which corresponds to option (b).

Step-by-step solution

Find the derivative of the function

To find the critical points, we need to first find the first derivative of the function with respect to x. The derivative of a sum is the sum of the derivatives, so we will differentiate each term separately: \[\frac{d}{dx}((x+a)^2) = 2(x+a)\] \[\frac{d}{dx}((x+b)^2) = 2(x+b)\] \[\frac{d}{dx}((x+c)^2) = 2(x+c)\] Now, we can find the derivative of the whole function: \[\frac{d}{dx}((x+a)^2 + (x+b)^2 + (x+c)^2) = 2(x+a) + 2(x+b) + 2(x+c)\]

Find the critical points

To find the critical points, we will set the derivative equal to zero and solve for x: \[2(x+a) + 2(x+b) + 2(x+c) = 0\] We can factor out the constant 2: \[2(x+a + x+b + x+c) = 0\] Now, we can simplify the expression and solve for x: \[2(3x + a + b + c) = 0\] Since the factor of 2 cannot be zero, we can divide both sides by 2: \[3x + a + b + c = 0\] Now, we can solve for x: \[x = -\frac{a + b + c}{3}\]

Verify if the critical point is a minimum

We will use the second derivative test to determine if the critical point is a minimum. First, we need the second derivative of the function: \[\frac{d^2}{dx^2}((x+a)^2 + (x+b)^2 + (x+c)^2) = 2 + 2 + 2 = 6\] Since the second derivative is always positive (6 in this case), the critical point is indeed a minimum.

Match the solution with the given options

From our analysis, we found that the minimum value of the given expression is at \(x = -\frac{a + b + c}{3}\). Looking at the given options, the correct choice would be: (b) \(\left[-(a+b+c) \right] / 3\)

Key concepts

Understanding Derivatives

In calculus, the derivative of a function provides us with essential information about its behavior. Specifically, it can tell us how the function's output changes as the input changes. When we're looking to minimize or maximize a function, the derivative helps us find those critical points where these changes are crucial.

In the given exercise, the derivative plays a vital role in identifying where the function achieves its minimum value. For the function \((x+a)^2 + (x+b)^2 + (x+c)^2\), our task starts by calculating the first derivative with respect to \(x\). This is done by differentiating each term separately, as derivatives add linearly:

• \(\frac{d}{dx}((x+a)^2) = 2(x+a)\)
• \(\frac{d}{dx}((x+b)^2) = 2(x+b)\)
• \(\frac{d}{dx}((x+c)^2) = 2(x+c)\)

Adding these, the derivative of the entire function becomes \(2(x+a) + 2(x+b) + 2(x+c)\). This expression helps us locate where the slope of the curve is zero, indicating potential minimum or maximum points.

Finding Critical Points

Once we have the derivative of a function, the next step is identifying the critical points. Critical points occur where the derivative equals zero or where the derivative doesn't exist. They are significant because they can indicate potential minimum or maximum values of the function.

In our example, we set the derivative \(2(x+a) + 2(x+b) + 2(x+c)\) equal to zero to find these points:

• \[2(x+a) + 2(x+b) + 2(x+c) = 0\]
• Factor out the constant 2: \[2 \times (3x + a + b + c) = 0\]
• Since 2 can't be zero, solve for \(3x + a + b + c = 0\)
• This simplifies to \[x = -\frac{a+b+c}{3}\]

Thus, the critical point for the function is \(x = -\frac{a+b+c}{3}\). This is where we need to verify further to determine if it truly represents a minimum.

Applying the Second Derivative Test

After finding our critical point, the second derivative test helps to determine whether this point is a minimum, maximum, or neither. The test involves finding the second derivative and evaluating its sign.

Here's how it works:

• If the second derivative at a critical point is positive, the function is concave up, indicating a local minimum.
• If it is negative, the function is concave down, pointing to a local maximum.
• If the second derivative is zero, the test is inconclusive.

In our exercise, we find the second derivative of the function \((x+a)^2 + (x+b)^2 + (x+c)^2\) to be

• \(\frac{d^2}{dx^2}\) = \(2 + 2 + 2 = 6\)

Since the second derivative is positive (specifically \(6\)), the function is indeed concave up at our critical point. This confirms that \(x = -\frac{a+b+c}{3}\) is where the function achieves its minimum, matching the options provided in the original exercise.