Question

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=-2 x^{2}+4 x+3 $$

Short answer

The critical number of the function \(f(x) = -2x^2 + 4x + 3\) is \(1\). The function increases over the interval \((-∞,1)\) and decreases over the interval \((1,∞)\).

Step-by-step solution

Find the first derivative

First, the derivative of the function \(f(x) = -2x^2 + 4x + 3\) must be computed. The derivative \(f'(x)\) can be given by \(-4x + 4\).

Determine the critical numbers

Critical numbers are found by setting the derivative equal to zero and solving for \(x\). For our function, we use the equation \(-4x + 4 = 0\). Solving it give us \(x = 1\). So, \(1\) is our critical number.

Determine the intervals of increase and decrease

The function increases for \( x < 1 \) and decreases for \( x > 1\). Hence, \((-∞,1)\) is our increasing interval and \((1,∞)\) is our decreasing interval.

Use a graphing utility to verify your results

Now, using a graphing tool (like desmos or geogebra), plot the function \(f(x) = -2x^2 + 4x + 3\). The plot confirms that the function is increasing from \(-∞\) to \(1\) and decreasing from \(1\) to \(∞\).

Key concepts

Understanding Critical Numbers

Finding the critical numbers of a function is akin to uncovering the 'turning points' where the function's graph changes direction. In calculus, these numbers are crucial for analyzing the behavior of functions, particularly regarding their increasing and decreasing trends.

For the function given, the critical numbers are the values of 'x' that make the first derivative—representing the slope of the tangent to the function's graph—equal to zero or undefined. For the quadratic function f(x) = -2x^2 + 4x + 3, the first derivative is straightforward and is found using basic differentiation rules: f'(x) = -4x + 4. Setting this derivative equal to zero, we are able to isolate the critical number as x = 1.

Understanding critical numbers can also lead to insights about where the function has local maxima or minima, which are essential in optimization problems.

Analyzing Increasing and Decreasing Intervals

Once critical numbers are identified, the next step is to analyze where the function is increasing or decreasing. This is pivotal as it reveals the 'ups' and 'downs' of the function's graph, providing a sketch of its overall shape without needing to plot it immediately.

For this, we create a number line and perform a 'test' on intervals around the critical numbers using the first derivative. If the derivative is positive in an interval, the function is increasing; if negative, decreasing. In our function, since the derivative—f'(x) = -4x + 4—is positive for x < 1 and negative for x > 1, we can clearly state that the function is increasing on the interval (-∞, 1) and decreasing on (1, ∞).

This analysis lets us truly understand the behavior of functions, aiding in drawing accurate graphs by hand and also in predicting the nature of the function's application in real-life scenarios.

Employing a Graphing Utility in Calculus

Graphing utilities, such as Desmos or GeoGebra, are more than just digital versions of traditional graph paper. They serve as vital tools in the field of calculus by providing visual representation and verification of our analytical findings.

By inputting the function f(x) = -2x^2 + 4x + 3 into a graphing calculator, we can visually confirm our calculations — the critical number and increasing and decreasing intervals. The graph will visibly show the function rising as it approaches x = 1 from the left and falling as it moves past x = 1 to the right, with the peak at x = 1 clearly depicting it as a local maximum.

Modern graphing utilities provide not only the graph but also features like zoom, trace, and analyze, which enable a deeper exploration of the function's properties. Leveraging these powerful utilities equips students with additional confirmation of their analytical skills and encourages further investigation into the intriguing world of calculus.