Second Derivative Analysis for End Behavior

00:00 5 marks

Consider the function \(f(x) = x^3 – 3x^2 + 4\). Use calculus to analyze its second derivative for large values of \(x\).

Part a

4 minutes 3 marks

Question

Differentiate \(f(x) = x^3 – 3x^2 + 4\) twice to obtain \(f”(x)\) and determine \(\lim_{x \to \infty} f”(x)\).

Hint Hint

Hint

Remember the power rule: \(\frac{d}{dx}x^n = nx^{n-1}\).

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Complete Answer

\(\lim_{x \to \infty} f”(x) = \infty\)

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Part b

3 minutes 2 marks

Question

Based on the result from part (a), determine the concavity of the function \(f(x)\) as \(x \to \infty\).

Hint Hint

Hint

A positive second derivative indicates concave upward behavior.

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Complete Answer

The function is concave upward for large \(x\).

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