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. $$y=2 x^{1 / 5}+3$$

Short answer

The graph of the function \(y = 2x^{1/5} + 3\) is concave up for \(x > 0\) and is concave down for \(x < 0\).

Step-by-step solution

Find the First Derivative

The derivative of the function \(y = 2x^{1/5} + 3\)is found by applying the power rule of differentiation. The power rule states: if \(y = x^n\), then \(y' = n*x^{(n-1)}\). Thus, the first derivative \(y'\) is found as follows: \[y' = 2 * (1/5) * x^{(1/5 - 1)} = 2/5 * x^{-4/5}.\] The term +3 vanishes as the derivative of a constant is zero.

Find the Second Derivative

The second derivative of the function is found by differentiating the derived first derivative from Step 1 again. Thus, applying the power rule once more to the first derivative, we find the second derivative \[y'' = 2/5 * (-4/5) * x^{(-4/5 - 1)} = -8 / (25*x^{9/5}).\]

Analyze the Sign of the Second Derivative to Determine the Concavity

The sign of the second derivative determines the intervals where the function is concave up and concave down. \[ y'' = -8 / (25*x^{9/5}) > 0 \] if \(x > 0\) and \[ y'' = -8 / (25*x^{9/5}) < 0 \] if \(x < 0\). Hence, the function is concave up for \(x > 0\) and concave down for \(x < 0\).

Key concepts

Concave Up Function

A function is considered to be concave up on an interval if the slope of its tangent lines is increasing over that interval. Think of it like a smiley face; it's the shape of a cup that can hold water. Mathematically, this relates to the function's second derivative.

For a function, if its second derivative is positive over a certain range of x-values, then the function is concave up on that interval. To put it into context using our example function, \( y = 2x^{1/5} + 3 \), we found that the second derivative \( y'' > 0 \) when \( x > 0 \). This tells us that for all positive values of x, the graph of \( y \) has a 'smiley face' curvature, or it's concave up.

Concave Down Function

On the flip side, a function is said to be concave down over an interval when the slope of its tangent lines decreases as x increases. Picture this as a frown, or an upside-down cup. Again, we refer to the second derivative to determine this property.

A function whose second derivative is negative over an interval will be concave down in that interval. Using our initial function, we derived the second derivative to be \( y'' = -8 / (25*x^{9/5}) \). From this, we can conclude that \( y'' < 0 \) when \( x < 0 \), indicating that the graph has a 'frown face' curvature, or it is concave down for all negative values of x.

Second Derivative Test

The second derivative test is a useful tool for determining the concavity of a function and the nature of its critical points. When you take the first derivative of a function and set it to zero, the solutions can identify potential maxima or minima, known as critical points.

The second derivative tells us more about these points. If the second derivative is positive at a critical point, it's a local minimum; if it's negative, the point is a local maximum. If it equals zero, the test is inconclusive, and other methods must be used to determine the point's nature.

For differentiable functions such as \( y = 2x^{1/5} + 3 \) where the second derivative is clearly negative or positive, this test gives a direct insight into the concavity of the graph, and additionally suggests where the graph changes its curvature, also known as points of inflection.

Power Rule of Differentiation

The power rule is a quick way to differentiate functions of the form \( y = ax^n \) where 'a' is a coefficient and 'n' is a real number power. According to this rule, the derivative of \( y \) with respect to 'x' is given by \( y' = nax^{n-1} \). This is a fundamental technique in calculus used for finding slopes and rates of change.

In the example \( y = 2x^{1/5} + 3 \) that we've been discussing, the power rule was used to find the first derivative, which came out to be \( y' = 2/5 * x^{-4/5} \) after the constant term dropped off since the derivative of a constant is zero. The power rule was then applied again to get the second derivative which played a critical role in the concavity test.