Question
Download Solution PDFWhat is the condition that f(x) = x3 + x2 + kx has no local extremum?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFFormula used:
\(\rm \frac{d}{dx} x^{n} = nx^{n - 1}\)
Calculation:
f(x) = x3 + x2 + kx
f'(x) = 3x2 + 2x + k
For no extremum, f should have zero turning points.
There will be no slope existing either it is +ve or -ve
For that above quadratic have no real roots.
⇒ f'(x) ≠ 0 which is only possible when,
22 - 4 × 3 × k < 0
⇒ 4 - 12k < 0
⇒ 4 < 12k
⇒ 1 < 3k
⇒ 3k > 1
∴ The condition that f(x) = x3 + x2 + kx has no local extremum is 3k > 1
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