The function \(y(x)\) is given by $$ y(x)=2 x^{3}-9 x^{2}+1 $$ (a) Calculate the values of \(x\) for which \(y^{\prime}=0\). (b) Calculate the values of \(x\) for which \(y^{\prime \prime}=0\). (c) State the interval(s) on which \(y\) is increasing. (d) State the interval(s) on which \(y\) is decreasing. (e) State the interval(s) on which \(y^{\prime}\) is increasing. (f) State the interval(s) on which \(y^{\prime}\) is decreasing.

The first derivative of a function, denoted as \( y'(x) \) or \( \frac{dy}{dx} \), provides critical information about the function's rate of change. In other words, it tells us how quickly or slowly the function value changes with respect to changes in \( x \). Calculating the first derivative is the first step in analyzing the behavior of functions. For our example function, \( y(x) = 2x^3 - 9x^2 + 1 \), taking the derivative using the power rule yields \( y'(x) = 6x^2 - 18x \).

Using the first derivative, we can determine where the function is increasing or decreasing. If the first derivative is positive over an interval, it indicates that the function is increasing there; conversely, a negative first derivative indicates a decreasing function. Setting \( y'(x) = 0 \) helps us find the critical points, which are used as boundaries to test these intervals.

The second derivative of a function, often expressed as \( y''(x) \) or \( \frac{d^2y}{dx^2} \), is the derivative of the first derivative. It provides insight into the concavity of the function—whether the function curves upwards or downwards. For the given function, we have \( y''(x) = 12x - 18 \).

Inflection points can be found where the second derivative changes sign, which occurs at values of \( x \) where \( y'' = 0 \) (in our case, \( x = \frac{3}{2} \)). These points are where the function switches from concave up to concave down, or vice versa. Moreover, when the second derivative is positive, the function is concave up, which is often associated with a 'smiley face' shape, and the function is concave down (like a 'frowny face') when the second derivative is negative.

To figure out where the function is increasing or decreasing, we look at the sign of the first derivative. If \( y'(x) > 0 \), the function \( y(x) \) is increasing; if \( y'(x) < 0 \), it is decreasing. For example, with our function's first derivative \( 6x^2 - 18x \), we have found that \( y(x) \) is increasing on the interval \( (0, 3) \) and decreasing on \( (-\text{fty}, 0) up (3, t\text{fty}) \).

This analysis is imperative for understanding the overall trend of the function across different regions of its domain, and it is especially useful for engineers in optimizing system behaviors or understanding the rates at which physical processes change.

Critical points of a function occur where its first derivative\( ygneq 0 \) is equal to zero, or when the derivative does not exist. For our function, setting the first derivative \( y'(x) \) to zero and solving for \( x \), yielding \( x = 0 \) or \( x = 3 \), gives us the function's critical points. These points are paramount in mapping the function's behavior as they represent locations where the function's rate of change is momentarily zero—implying either a local maximum, minimum, or a saddle point (neither).

Identifying and analyzing critical points is crucial for engineers when they endeavor to find stress points in structures, determine the most efficient operating parameters, or any other scenario where extremal values are of interest.

Inflection points are points on the graph of a function at which the curvature changes sign. They are identified by setting the second derivative equal to zero and solving for \( x \) or by points where the second derivative is undefined but the function is still continuous. For our function, we determined an inflection point at \( x = \frac{3}{2} \).

Inflection points are practical for understanding the geometry of functions and can signal a change in the nature of extremal values from a maximum to a minimum or vice versa. This understanding is particularly essential for engineers who need to optimize design and functionality while considering the stiffness and flexibility of materials.