Problem 25 - Essential Calculus

Question

Find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=x^{2 / 3}-3\)

Step-by-Step Solution

Answer

There are no relative extrema for this function.

Step 1: Find the First Derivative

The first derivative of the function \(f(x) = x^{2/3} - 3\) is calculated using power rule in derivation. The derivative of this function \(f'(x)= \frac{2}{3}x^{-1/3}\). This derivative will be used to find the critical numbers.

Step 2: Find the Critical Numbers

Critical numbers are values of x where the first derivative is either zero or undefined. By setting the first derivative to zero and solving for x, \( \frac{2}{3}x^{-1/3} = 0\), we find no solution. However, when we check for where the derivative is undefined, it occurs at x=0. So, the critical number is x=0.

Step 3: Find the Second Derivative

The second derivative of the function is calculated by differentiating the first derivative. The second derivative \(f''(x)= \frac{-2}{9}x^{-4/3}\). This will be used in the Second Derivative Test to determine if the critical numbers are relative minimum, maximum or neither.

Step 4: Use The Second Derivative Test

The Second Derivative Test states that if a function's second derivative at a point is positive, then the function has a relative minimum at that point. If it's negative, then the function has a relative maximum. If it equals zero, the test is inconclusive. Plugging in the critical number (0) into the second derivative, we get an undefined value, hence the Second Derivative Test is inconclusive at x=0.

Step 5: Find the Y-coordinate of the Extrema

Because the Second Derivative Test is inconclusive, we must look to the original function to determine the y-coordinate corresponding to the extrema. Substituting the x-coordinate (0) into the original function we get \(f(0) = 0^{2/3} - 3 = -3\). Hence, what we have is a point on the curve, but it’s neither a maximum or a minimum.