JEE Main & Advanced Mathematics Applications of Derivatives Question Bank

done Numerical Value Type Questions - Monotonicity Total Questions - 10

• question_answer1)
  Function \[f\left( x \right)={{x}^{3}}+6{{x}^{2}}+\left( 9+2k \right)x+1\,\,\forall \,\,x\in R\] is strictly increasing function if \[k>\lambda \] then find \[\lambda .\]
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• question_answer2)
  The function \[f\left( x \right)=\frac{a\,\sin x+2\cos x}{\sin x+\cos x}\] is increasing for \[x\in R\] when \[a>\lambda \] then find \[\lambda \].
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• question_answer3)
  Let \[\phi \left( x \right)=f\left( x \right)+f\left( 2a-x \right)\] and \[f''\left( x \right)>0\,\,;\,\,\forall \,\,x\in \left[ 0,a \right].\] If \[\phi \left( x \right)\] is decreasing in \[\left[ 0,\lambda a \right]\] then find \[\lambda \].
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• question_answer4)
  Function \[f\left( x \right)=2{{x}^{3}}-9{{x}^{2}}+12x+29\] is decreasing when \[x\in \left( a,b \right)\] then find value of \[a+b.\]
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• question_answer5)
  The interval in which \[2{{x}^{3}}+5\] increases less rapidly than \[9{{x}^{2}}-12x,\] is \[\left( a,b \right)\] then find \[a+b.\]
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• question_answer6)
  If the function f: \[R\to R\] given by \[f\left( x \right)={{x}^{3}}+a{{x}^{2}}+5x+\sin \,\,2x\] is invertible and \[a\in \left( -3,1 \right)\] then find \[\lambda \].
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• question_answer7)
  If \[f\left( x \right)={{x}^{3}}+4{{x}^{2}}+\lambda x+1\] is monotonically decreasing function of x in the largest possible interval \[\left( -2,-2/3 \right);\] then find \[\lambda \].
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• question_answer8)
  If the complete set of values of x in which \[f\left( x \right)=2{{\log }_{e}}\left( x-2 \right)-{{x}^{2}}+4x+1\] increases, is \[\left( a+b \right)\] then find \[a\text{ }+\text{ }b\].
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• question_answer9)
  If \[f\left( x \right)=k{{x}^{3}}-9{{x}^{2}}+9x+3\] is monotonically increasing \[\forall \,\,x\in \,R\] and \[k\in \left[ \lambda ,\left. \infty  \right), \right.\] then find value of\[\lambda \].
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• question_answer10)
  If \[f\left( x \right)=x-2\sin x,0\,\,\le \,x\le 2\pi \] is increasing in the interval \[\left[ a\pi ,b\pi  \right],\] then find value of \[a+b.\]
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