COMEDK · Maths · 28. Indefinite Integration
Given that \(f(x)=k x^3-9 k x^2+9 x+3\) is an increasing function on R set of real numbers, then \(k\) belongs to
- A \(\left(0, \dfrac{1}{3}\right]\)
- B \(\left(-\dfrac{1}{3}, 0\right)\)
- C \(\left[\dfrac{1}{3}, \infty\right)\)
- D \((-\infty, 0)\)
Answer & Solution
Correct Answer
(A) \(\left(0, \dfrac{1}{3}\right]\)
Step-by-step Solution
Detailed explanation
The function \(f(x) = kx^3 - 9kx^2 + 9x + 3\) is increasing on \(\mathbb{R}\) if and only if its derivative \(f'(x) \ge 0\) for all \(x \in \mathbb{R}\). Calculating the derivative: \(f'(x) = 3kx^2 - 18kx + 9\). For \(f'(x) \ge 0\) for all \(x\), the quadratic expression…
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