91Ó°ÊÓ

Critical points occur where the derivative of a function equals zero or is undefined. These points are essential in finding the extrema, or the highest and lowest values, of the function within a given interval.

• To identify critical points, calculate the derivative of the function.
• For our function, we found the derivative to be: \(f'(x) = 4x^3 - 10x\).
• Solve the equation \(4x^3 - 10x = 0\) to find where the derivative is zero.

This equation simplifies to \(2x(2x^2 - 5) = 0\), giving the potential solutions \(x = 0\) and \(x = \pm\sqrt{\frac{5}{2}}\).
Since our interval is \([0, 2]\), only include solutions within this range. Hence, we consider \(x = 0\) and \(x = \sqrt{\frac{5}{2}}\). Critical points are important as they highlight where the function's slope changes, possibly indicating a local minimum or maximum.

The derivative is a function that represents the rate of change of one variable with respect to another. It can be thought of as the slope of the tangent line to the graph of a function.
For functions like \(f(x) = x^4 - 5x^2 + 4\), the derivative helps us understand how the function changes as \(x\) changes. Calculating the derivative involves applying rules of differentiation:

• For any term \(x^n\), the derivative is \(nx^{n-1}\).
• Thus, the derivative of \(x^4\) is \(4x^3\) and of \(-5x^2\) is \(-10x\).

The resulting derivative, \(4x^3 - 10x\), helps us locate critical points by setting it to zero. This operation provides the values of \(x\) where the function has either a horizontal tangent or potentially reaches its extremum.

An interval specifies the range of values, often for the variable \(x\), under consideration. In this problem, keeping the interval in mind is crucial to identifying relevant critical points and determining the function's extrema.
The interval from the problem statement, \([0, 2]\), provides boundaries for where you should evaluate the function. Here's how it affects the process:

• Solutions for critical points outside the interval, like negative roots, are not considered.
• The endpoints, \(x = 0\) and \(x = 2\), also need to be evaluated as they may be points of extrema.

After determining the critical points, evaluate the function at these points and include the endpoints. Compare these values to establish local minima or maxima within the interval. This step ensures you consider the complete behavior of the function within the specified range, leading to correctly identifying any extremum present in \([0, 2]\).