TS EAMCET · Maths · Application of Derivatives
Consider the function \(f(x)=2 x^3-3 x^2-x+1\) and the intervals \(Y_1=[-1,0], Y_2=[0,1]\), \(r_3=[1,2], r_4=[-2,-1]\). Then,
- A \(f(x)=0\) has a root in the intervals \(f_1\) and \(f_4\) only
- B \(f(x)=\) ohas a root in the intervals \(f_1\) and \(f_2\) only
- C \(f(x)=0\) has a root in every interval except in \(f_4\)
- D \(f(x)=0\) has a root in all the four given intervals
Answer & Solution
Correct Answer
(C) \(f(x)=0\) has a root in every interval except in \(f_4\)
Step-by-step Solution
Detailed explanation
We have, \[ f(x)=2 x^3-3 x^2-x+1 \] Let \[ \begin{aligned} & g(x)=\frac{x^4}{2}-x^3-\frac{x^2}{2}+x \\ & g(-1)=\frac{1}{2}+1-\frac{1}{2}-1=0 \end{aligned} \] and \[ g(0)=0 \] \(\therefore f(x)=0\) has roots lie in \([-1,0]\). Similarly, \(g(0)=g(1)=0\)…
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