Question

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=(x-1)^{5}\)

Short answer

The graph of the function \(y=(x-1)^{5}\) reveals a point of inflection at \(x=1\) with no visible local maximum or minimum. The function is increasing on \((-∞, 1)\) and \((1, ∞)\).

Step-by-step solution

Identify the roots of the function

For the function \(y=(x-1)^{5}\), the root is where the function crosses the x-axis. This happens when \(y=0\). Therefore, setting the function to zero, we get \(0=(x-1)^{5}\) which gives \(x=1\) as the root.

Find the first derivative and identify extrema

The first derivative of the function tells us about its slope. A point at which the derivative is zero can be a maximum, minimum, or a point of inflection. The first derivative is \(\frac{dy}{dx}=5(x-1)^{4}\). Setting \(\frac{dy}{dx}=0\) gives \(0=5(x-1)^{4}\) which only satisfies for \(x=1\). This suggests that there might be a point of inflection or extrema at \(x=1\). However, since the power 5 is an odd number, our function will not have any maximum or minimum point.

Find the second derivative

The second derivative of a function shows where the function's slope changes direction, which is where points of inflection occur. The second derivative is \(\frac{d^2y}{dx^2}=20(x-1)^{3}\).

Identify points of inflection

Setting the second derivative equal to zero, \(\frac{d^2y}{dx^2}=0\), we obtain \(0=20(x-1)^{3}\) which again only satisfies for \(x=1\). Therefore, the function has a point of inflection at \(x=1\). Since there is no local minimum or maximum points, the inflection point is where the function changes concavity.

Sketch the graph

Knowing these points, we can sketch a rough graph of the function \(y=(x-1)^{5}\). Since the function is odd, it will go towards negative infinity as x approaches negative infinity and towards positive infinity as x approaches positive infinity. It should transition smoothly with a point of inflection at \(x=1\) and no visible local extrema.