TS EAMCET · Maths · Application of Derivatives
If \(f(x)=k x^3-3 x^2-12 x+8\) is strictly decreasing for all \(x \in \mathbf{R}\) then
- A \(k \lt -\frac{1}{4}\)
- B \(k\gt-\frac{1}{4}\)
- C \(k\gt\frac{1}{4}\)
- D \(k \lt \frac{1}{4}\)
Answer & Solution
Correct Answer
(A) \(k \lt -\frac{1}{4}\)
Step-by-step Solution
Detailed explanation
\(f(x)=k x^3-3 x^2-12 x+8\) \(f^{\prime}(x)=3 k x^2-6 x-12 \lt 0 \forall x \in R\) This implies \(3 k \lt 0 \Rightarrow k \lt 0\) \(\begin{aligned} & \Delta \lt 0 ; 36+144 k \lt 0 \\ & k \lt \frac{-1}{4} \quad \therefore k \lt \frac{-1}{4} \end{aligned}\)
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