Question

In Exercises \(7-12\) , use the Concavity Test to determine the intervals on which the graph of the function is (a) concave up and (b) concave down. $$y=5-x^{1 / 3}$$

Short answer

The function \(y = 5 - x^{1/3}\) is concave up where \(x > 0\) and concave down where \(x < 0\). Concavity is undefined at \(x = 0\).

Step-by-step solution

Calculate the first derivative

For the function \(y = 5 - x^{1/3}\), the first derivative \(y'\) is calculated using the power rule for derivatives: \(-\frac{1}{3}x^{-2/3}\). This simplifies to \(-\frac{1}{3x^{2/3}}\).

Determine the second derivative

The second derivative \(y''\) is calculated by differentiating the first derivative \(-\frac{1}{3x^{2/3}}\). The result is \(\frac{2}{9x^{5/3}}\).

Check the sign of the second derivative

The function is concave up where the second derivative is positive, and concave down where the second derivative is negative. Since \(x^{5/3}\) is always positive for any real number x, the second derivative \(\frac{2}{9x^{5/3}}\) is also always positive as long as \(x \neq 0\). Therefore, the function is concave up where \(x > 0\) and concave down where \(x < 0\).

Examine at the point of discontinuity

No continuity exists at \(x=0\) (since the denominator becomes zero in the second derivative). Hence, no conclusion regarding concavity can be made at this point.