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

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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