Question

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant. $$f(t)=\frac{1}{5} t^{5}-a^{4} t$$

Short answer

Answer: The critical points of the function are at t = a and t = -a.

Step-by-step solution

Find the derivative of the function

To find the critical points, we have to find the derivative of the given function with respect to t. The function is: $$f(t)=\frac{1}{5}t^{5}-a^{4}t$$ Using the power rule to differentiate, we get: $$f'(t) = \frac{d}{dt} (\frac{1}{5}t^{5}-a^{4}t)$$ $$f'(t) = \frac{1}{5}(5t^{4}-a^{4})$$

Set the derivative equal to zero

To find the critical points, set the derivative equal to zero and solve for t: $$f'(t) = t^{4}-a^{4} = 0$$

Solve for t

Now, we will solve the above equation to find the values of t: $$t^{4} = a^{4}$$ Taking the fourth root of both sides, we get: $$t = \pm a$$

Identify the critical points

As a result, the critical points of the function f(t) are at t = a and t = -a.

Key concepts

Derivative of a Function

Understanding the derivative of a function is essential in calculus. It represents the rate at which a function's value changes at any given point. To find the derivative, we use rules of differentiation, which are methods to calculate the slope of the tangent line to the function's graph at any point. This slope is indicative of how fast the function is increasing or decreasing at that point.

For the function \(f(t) = \frac{1}{5}t^{5} - a^{4}t\), its derivative \(f'(t)\) signifies how \(f(t)\) changes with respect to \(t\). In the context of critical point analysis, the derivative tells us where the function levels out or where it stops increasing or decreasing and starts doing the opposite. These flat points or turning points can potentially be maximums, minimums, or inflection points on the function's curve.

Power Rule Differentiation

The power rule is a fundamental technique in differentiation used for finding the derivative of a function that involves a power of \(t\). The rule states that if \(f(t) = t^{n}\) for a real number \(n\), then \(f'(t) = nt^{n-1}\). This is a quick way to differentiate functions without the need for more complex methods.

In our exercise, we apply the power rule to each term individually. For the term \(\frac{1}{5}t^5\), we multiply the exponent by the coefficient (5 times \frac{1}{5}), and then subtract one from the exponent to get \(t^4\). Therefore, \(f'(t) = t^4 - a^4\) when applying the power rule to the original function \(f(t)\). This simplicity and efficiency are what make the power rule a go-to method for finding derivatives, especially when dealing with polynomial functions.

Solving Equations

Solving equations is a process to find the value of the unknown variable that makes the equation true. In finding critical points, we commonly set the function's derivative equal to zero and solve for the variable to determine at which points the function doesn't increase or decrease. In our example, setting \(f'(t) = t^4 - a^4 = 0\) is crucial as it leads us to the values of \(t\) where the function's rate of change is zero.

Solving \(t^4 = a^4\) involves taking the fourth root of both sides, resulting in \(t = \[ \-a, a \]\). This yields the critical points of the function. It's important to check all potential solutions in the original equation, as some might not be valid in the given context or might lead to undefined values within the function's domain.