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Explain solution rd sharma class 12 chapter 16 Increasing and decreasing function exercise multiple choice question, question 29

Answers (1)

Correct option (a)

Hint: Iff(x)  is increasing function {f}'(x)>0

Given:f(x)=x^{3}-9 k x^{2}+27 x+30

Explanation:It is given that

f(x)=x^{3}-9 k x^{2}+27 x+30      …..(i)

Differentiate (i) w.r.t  x

\begin{aligned} &f^{\prime}(x)=3 x^{2}-9 k \cdot 2 x+27 \\ &f^{\prime}(x)=3\left(x^{2}-6 k x+9\right) \end{aligned} 

\because f(x) is increasing

\begin{aligned} &3\left(x^{2}-6 k x+9\right)>0 \\ &x^{2}-6 k x+9>0 \end{aligned}

In a x^{2}+b x+c=0 , if a>0  thenb^{2}-4 a c<0

\begin{aligned} &\Rightarrow(-6 k)^{2}-4 \times 1 \times 9<0 \\ &36 k^{2}-36<0 \\ &k^{2}-1<0 \\ &(k+1)(k-1)<0 \\ &\Rightarrow-1<k<1 \end{aligned} 

Thus , if f(x)  is increasing function then  -1\leq k< 1  

Note:- option (a) has to be  -1\leq k< 1

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