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The first derivative of a function is a powerful tool in calculus, used to find the critical points of a function, which can indicate where local maxima or minima might occur. For a function like \(H(x) = x^4 - 2x^3\), the first derivative \(H'(x)\) helps us locate where the slope of the tangent line is zero, which corresponds to potential turning points in the function.

To compute the first derivative of \(H(x) = x^4 - 2x^3\), we differentiate term by term:

• The derivative of \(x^4\) is \(4x^3\).
• The derivative of \(-2x^3\) is \(-6x^2\).

Thus, \(H'(x) = 4x^3 - 6x^2\).

Setting the derivative \(H'(x) = 0\) lets us solve for the \(x\) values where the slope is zero:

• Factor: \(2x^2(2x - 3) = 0\).
• Solve: \(x = 0\) or \(x = \frac{3}{2}\).

Once critical points are found using the first derivative, the second derivative helps to ascertain the nature of these points: whether they are points of local maxima, minima, or points of inflection. The second derivative test gives us more information on the concavity of the function at these critical points.

For \(H(x) = x^4 - 2x^3\), the second derivative \(H''(x)\) is found by differentiating \(H'(x)\):

• The derivative of \(4x^3\) is \(12x^2\).
• The derivative of \(-6x^2\) is \(-12x\).

Thus, \(H''(x) = 12x^2 - 12x\).

Checking the concavity of the function at the critical points involves substituting these \(x\) values into \(H''(x)\):

• If \(H''(x) > 0\), the function is concave up, indicating a local minimum.
• If \(H''(x) < 0\), the function is concave down, indicating a local maximum.
• If \(H''(x) = 0\), further analysis is required as it may indicate a point of inflection.

A local maximum is a point where a function reaches a peak in a given interval, meaning no nearby points have a greater value. To identify a local maximum, we use the first and second derivatives of a function. However, in this particular example involving \(H(x) = x^4 - 2x^3\), a local maximum was not present among the tested critical points.

When exploring critical points derived from \(H'(x) = 0\), the second derivative did not reveal any concave down areas (where \(H''(x) < 0\)) at these specific points, which would suggest a local maximum.

Nonetheless, identifying local maxima is crucial when analyzing functions as it allows us to comprehend peak behaviors and optimize or minimize values accordingly across various applications.

A local minimum is the opposite of a local maximum, representing a point in an interval where the function's value is less than at any nearby points. For the function \(H(x) = x^4 - 2x^3\), we employed the critical points identified through \(H'(x) = 0\) to evaluate for a local minimum.

By using the second derivative test on the critical point \(x = \frac{3}{2}\), we found \(H''\left(\frac{3}{2}\right) = 9\), which is positive. This positivity indicates that the function is concave up, confirming the presence of a local minimum at this point.

Additionally, we computed the actual value at this local minimum by plugging \(x = \frac{3}{2}\) into the original function \(H(x)\):

• \(H\left(\frac{3}{2}\right) = \left(\frac{3}{2}\right)^4 - 2\left(\frac{3}{2}\right)^3 = -\frac{27}{16}\).

This calculation verifies both the minimum location and its corresponding value.