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If the slope of tangent to the curve y = ax3 – bx2 at x = 1 is equal to 3, where a ∈  [1,2] then number of possible integral values of b is

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    Answered by

    Vishal Baghel | Contributor-Level 10

    a month ago

    d y d x = 3 a x 2 − 2 b x

    d y d x | x = 1 = 3 a − 2 b = 3

    a = 2 b + 3 3 ∈ [ 1 , 2 ]

    2 b + 3 ∈ [ 3 , 6 ]

    2 b ∈ [ 0 , 3 ]

    b ∈ [ 0 , 3 2 ]

Similar Questions for you

Curve y = 2x – x2, y(x) & y′(x) cut x-axis in M & N number of points respectively, find M + N.
A
alok kumar singh

y (x) = 2x – x2

y? (x) = 2x log 2 – 2x

M = 3

N = 2

M + N = 5

If the tangent to the curve y = x3 at the point P(t, t3) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1:2 is:
V
Vishal Baghel

y = x3

d y d x = 3 x 2 ⇒ d y d x | ( t , t 3 ) = 3 t 2

Equation of tangent y – t3 = 3t2 (x – t) 

Let again meet the curve at Q ( t 1 , t 1 3 )

⇒ t 1 3 − t 3 = 3 t 2 ( t 1 − t )

t 1 2 + t t 1 + t 2 = 3 t 2 [ ? t 1 ≠ t ]

t 1 2 + t t 1 − 2 t 2 = 0

=> t1 = -2t

Required ordinate = 2 t 3 + t 1 3 3 = 2 t 3 − 8 t 3 3 = − 2 t 3   

For x > 0, if f(x) = ∫ 1 x l o g e t ( 1 + t ) d t ,  then f(e) + f ( 1 e )  is equal to:
A
alok kumar singh

Given f(X) = ∫ 1 x l o g e t ( 1 + t ) d t . . . . . . . . . . . . ( i )  

So   f ( 1 x ) = ∫ 1 1 / x λ l o g e t 1 + t d t . . . . . . . . . . . . ( i i )

put t = 1 z       t h e n     d t = − 1 z 2 d z  

f ( 1 x ) = ∫ 1 x l o g e t t ( 1 + t ) d t . . . . . . . . . . . . ( i i i )

(i) + (iii), f(x) + f ( 1 x ) = ∫ 1 x ( l o g e t 1 + t + l o g e t t ( 1 + t ) ) d t

= [ ( l o g e t ) 2 2 ] 1 x = ( l o g x x ) 2 2

Hence f(e) + f ( 1 e ) = 1 2

Set of all possible values of k for which f(x) = sin x – cos x – kx + b decreases for all real values of x, is:
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Vishal Baghel

f' (x) = cosx + sinx − k ≤ 0∀x ∈ R
k ≥ √2

The graph of function f(x) = x5/20 - x4/12 + 5 has :
V
Vishal Baghel

f (x) = x? /20 - x? /12 + 5
f' (x) = x? /4 - x³/3 = x³ (x/4 - 1/3)
Local maxima at 0, Local minima at 4/3
f' (x) = x³ - x² = x² (x-1)
x = 1 point of inflection

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