Problem 5 - 5 Steps to A 5: AP Calculus AB 2021

Question

Show that the absolute minimum of \(f(x)=\sqrt{25-x^{2}}\) on [-5,5] is 0 and the absolute maximum is 5.

Step-by-Step Solution

Answer

Answer: The absolute minimum value is 0 and the absolute maximum value is 5.

Step 1: Find the first derivative of f(x)

First, we differentiate the function with respect to x. Since \(f(x) = \sqrt{25 - x^2}\), we can rewrite this as \(f(x) = (25 - x^2)^{1/2}\). Then, applying the chain rule to find the derivative: $$f'(x) = \frac{1}{2}(25 - x^2)^{-1/2}(-2x) = \frac{-x}{\sqrt{25 - x^2}}$$

Step 2: Determine the critical points

Now we must find the critical points of the function, so we must find where \(f'(x) = 0\) or \(f'(x)\) is undefined. The first derivative is undefined when the denominator is zero, which happens when x = ±5. So, let's find where the first derivative equals zero: $$\frac{-x}{\sqrt{25 - x^2}} = 0$$ Multiplying both sides of the equation by \(\sqrt{25 - x^2}\), we get: $$-x = 0$$ So, we have one critical point at \(x=0\).

Step 3: Evaluate the function at critical points and endpoints

We will now evaluate the function \(f(x)\) at all critical points and the two endpoints (-5 and 5) since the function is continuous everywhere in \([-5, 5]\). The two critical points are x = -5, 0, and 5. $$f(-5) = \sqrt{25 - (-5)^2} = \sqrt{0} = 0$$ $$f(0) = \sqrt{25 - 0^2} = \sqrt{25} = 5$$ $$f(5) = \sqrt{25 - 5^2} = \sqrt{0} = 0$$

Step 4: Determine the absolute minimum and maximum

Comparing the values of the function, we see that: - Absolute minimum is 0, which occurs at x = -5 and x = 5. - Absolute maximum is 5, which occurs at x = 0. Hence, the absolute minimum of \(f(x)\) on the interval \([-5, 5]\) is 0, and the absolute maximum is 5.

Key Concepts

Critical Points

Critical points are specific values in the domain of a function where its derivative is either zero or undefined. These points are essential for finding the local maxima and minima, as well as identifying potential locations of absolute extrema on a given interval.

• To find critical points, follow these steps: Calculate the derivative of the function.
• Solve the equation where the derivative is zero.
• Identify points where the derivative is undefined.

For the function \(f(x) = \sqrt{25 - x^2}\), the derivative is \(f'(x) = \frac{-x}{\sqrt{25 - x^2}}\). Solving \(f'(x) = 0\) gives us a critical point at \(x = 0\). Since the expression is undefined when the denominator \(\sqrt{25 - x^2}\) is zero, the points \(x = -5\) and \(x = 5\) are also critical.

Absolute Extrema

Absolute extrema refer to the absolute maximum or minimum values that a function attains on a specific interval. Unlike local extrema, which might occur in smaller open intervals within a function's domain, absolute extrema evaluate the function across a broader interval and include endpoints.

• To determine absolute extrema:
• Evaluate the function at each critical point and at the endpoints of the interval.
• Compare these values to find the greatest and smallest ones.

For \(f(x) = \sqrt{25 - x^2}\) over \([-5, 5]\), we evaluated \(f(-5) = 0\), \(f(0) = 5\), and \(f(5) = 0\). The absolute maximum value is 5, and the absolute minimum is 0.

Differentiation

Differentiation is the process of finding a derivative, which represents the rate of change of a function with respect to its variable. Derivatives help identify critical points and can also be useful in optimizing real-world scenarios where you need to find maximum or minimum values.

• A derivative can be found using basic rules like the power rule, product rule, quotient rule, or the chain rule.
• For composite functions, the chain rule is particularly useful.

In this exercise, differentiating \(f(x) = (25 - x^2)^{1/2}\) gave us \(f'(x) = \frac{-x}{\sqrt{25 - x^2}}\), using the chain rule due to the composition of a square root function.

Chain Rule

The chain rule is a technique in calculus used to differentiate composite functions. It's like a two-step process that allows you to deal with functions nested within one another.

• The general formula for the chain rule is: if you have a composite function, \(y = g(f(x))\), then its derivative is \(y' = g'(f(x)) \cdot f'(x)\).
• This approach simplifies differentiation tasks that might otherwise be very complex.

In this specific problem, the function \(f(x) = (25 - x^2)^{1/2}\) is a composite function. By applying the chain rule, we first differentiated the outer function, \((...)^{1/2}\), then multiplied it by the derivative of the inner function, \((25 - x^2)\). This led us to the derivative \(f'(x) = \frac{-x}{\sqrt{25 - x^2}}\).