Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(x=c\) is a critical number of the function \(f,\) then it is also a critical number of the function \(g(x)=f(x)+k,\) where \(k\) is a constant

Short answer

The statement is True. Any critical number for \(f(x)\) will remain a critical number for \(g(x) = f(x) + k\) where \(k\) is a constant, regardless of the value of \(k\).

Step-by-step solution

Understand the problem

The function \(g(x) = f(x) + k\) is basically the function \(f(x)\) shifted vertically by the constant \(k\). A critical number of a function is a number at which the derivative of the function is either zero or does not exist. Here, we need to determine whether a critical number \(x = c\) of \(f(x)\) will remain a critical number for the function \(g(x)\).

Analyze the derivative of the function \(g(x)

Considering the function \(g(x) = f(x) + k\), find the derivative \(g'(x)\). The derivative of \(g(x)\) would be \(g'(x) = f'(x) + 0\) (since the derivative of a constant is zero). This implies that the derivative of \(g(x)\) is the same as the derivative of \(f(x)\), regardless of the value of \(k\).

Connect the analysis to the problem

Since \(x = c\) is a critical number of \(f(x)\), then by definition, \(f'(c) = 0\) or does not exist. Thus considering our result in Step 2, \(g'(c)\) would also be 0 or undefined. This means that \(x = c\) is also a critical point of \(g(x)\).

Key concepts

Critical Points Definition

Critical points of a function are pivotal in the study of calculus, dictating where a function's derivative is zero or does not exist. Imagine you're climbing a mountain; critical points symbolize spots where you stop ascending or descending — the peaks, valleys, or plateaus. Mathematically, for a function f(x), a critical point occurs at x = c if the derivative f'(x) is zero (f'(c) = 0) or undefined. Critical points are essential as they help identify extremum points (maximums and minimums) and inflection points, where the function changes curvature.

To find these critical points, one must first calculate the derivative of the function and examine where it equals zero or is not defined. It's a bit like looking at a speedometer; when the speed drops to zero or the speedometer breaks, you've hit a critical point in your journey – that's what we look for in the graph of a function. Understanding critical points is crucial since they offer valuable insights into a function's behavior and help in solving optimization problems in various fields like economics, engineering, and beyond.

Derivative of a Function

The derivative of a function is a fundamental concept, akin to the instant snapshot of something in motion. Imagine a function as a smooth path and its derivative as the function's speed at any point along that path. For a function f(x), its derivative f'(x) signifies the rate at which f(x) is changing at a particular point. If you were driving, the derivative would be your speedometer reading at any given moment.

To compute the derivative, we employ calculus tools such as the power rule, product rule, quotient rule, and chain rule. These tools help us to differentiate functions and uncover the precise spots, the critical points, where the function's rate of change is nil or instantaneous changes are not definable. It's like discovering at what points along your path you halt or the path becomes infinitely steep. Therefore, understanding how to derive a function is essential for analyzing the function's growth rate, finding tangents to curves, and many more applications in both theoretical and practical problems.

Vertical Shifts in Functions

Vertical shifts are one of the simplest transformations we can apply to functions. They occur when we add or subtract a constant value k to the function f(x), resulting in a new function g(x) = f(x) + k. Visualize it as lifting or pushing down the entire graph of f(x) without altering its shape. It's like moving a picture up or down on a wall – the picture stays the same, but its position changes.

Importantly, vertical shifts do not affect the function's derivative other than translating it up or down along the y-axis. Therefore, if x = c is a critical point of f(x), then it will also be a critical point of the shifted function g(x), since the derivative at that point remains unchanged apart from the shift. In practice, this transformation is used to adjust models to fit data without changing the model's intrinsic properties, much like tweaking the elevation of a slide in a playground while keeping its shape intact for the same sliding experience.