Questions: For the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither; b) identify intervals where the function is increasing or decreasing; c) give the coordinates of any points of inflection; d) identify intervals where the function is concave up or concave down, and e) sketch the graph. h(x)=4 x^3-48 x

For the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither; b) identify intervals where the function is increasing or decreasing; c) give the coordinates of any points of inflection; d) identify intervals where the function is concave up or concave down, and e) sketch the graph.

h(x)=4 x^3-48 x

Transcript text: For the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither; b) identify intervals where the function is increasing or decreasing; c) give the coordinates of any points of inflection; d) identify intervals where the function is concave up or concave down, and e) sketch the graph. \[ h(x)=4 x^{3}-48 x \]

Solution

Solution Steps

Step 1: Find the derivative and critical points

The first derivative is \(f'(x) = 12 x^{2} - 48\). Solving \(f'(x) = 0\) gives critical points at -2, 2.

Step 2: Classify the critical points

At \(x = -2\), the function is concave down (relative maximum). At \(x = 2\), the function is concave up (relative minimum).

Step 3: Determine intervals of increase or decrease

In the interval (-oo, 0), the function is increasing. In the interval (0, oo), the function is decreasing.

Step 4: Find points of inflection

Points of inflection are found at 0.

Step 5: Determine intervals of concavity

In the interval (-oo, oo), the function is concave down.

Final Answer:

The analysis of the cubic function includes the classification of critical points, intervals of increase or decrease, points of inflection, and intervals of concavity. For detailed results, refer to the steps above.

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