Question

The set of all values of a for which the function

                   

decreases for all real x is

Solution

Correct option is

Differentiating, we get

    

For f (x) to be decreasing for all x, we must have f ‘(x) < 0 for all x.

  

This is possible only if

         

This inequality is always true if a > 1, i.e., a ∈ (1, ∞). Moreover, we must have a ≥ – 4 for  to be real. Therefore, we have

          

[∴  we consider only a < 1]      

⇒ a + 4 ≤ 1 + a2 – 2a ⇒ 0 ≤ a2 – 3a – 3     

   

SIMILAR QUESTIONS

Q1

The coordinates of points P(xy) lying in the first quadrant on the ellipsex2/8 + y2/18 = 1 so that the area of the triangle formed by the tangent at Pand the coordinate axes is the smallest, are given by

Q2

The points(s) on the curve y3 + 3x2 = 12y where the tangent is vertical is(are)

Q3

The equation of the common tangent to the curves y2 = 8x and xy = –1 is 

 

Q4

If ab > 0 then the minimum value of  

Q5

The curve y = ax3 + bx2 + cx + 8 touches x – axis at P(2, 0) and cuts they – axis at a point Q where its gradient is 3. The value of a, b, c are respectively

Q6

If the tangent at (1, 1) on y2 = x(2 – x)2 meets the curve again at P, is

Q7

The tangent to the curve 

At the point corresponding to  is

Q8

The points of contact of the vertical tangents to x = 2 – 3 sinθ,  y = 3 + 2 cos θ are

Q9

 the in this interval

Q10