MCQ (Previous Year Questions) - Maxima And Minima (Competition Level 1)

If the function f(x) = 2x³ – 9ax² + 12a² x + 1, where a > 0, attains its maximum and minimum at p and q respectively such that p² = q, then a equals

[AIEEE 2003]

The real number x when added to its inverse gives the minimum value of the sum at x equal to

[AIEEE 2003]

The function f(x)  has a local minimum at –

[AIEEE 2006]

A  triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length x. The maximum area enclosed by the park is –

[AIEEE 2006]

If p and q are positive real numbers such that  p² + q² = 1, then the maximum value of (p + q) is

[AIEEE 2007]

Suppose the cubic x³ – px + q has three distinct real roots where p > 0 and q > 0. Then which one of the following holds ?    

[AIEEE 2008]

Given P(x) = x⁴ +ax³ + bx² +cx + d such that x = 0 is the only real root of P’ (x) =0. If P(–1) < P(1), then in the interval [–1,1] -

[AIEEE 2009]

The shortest distance between the line y – x = 1 and the curve x = y² is -

[AIEEE 2009]

Let f : R → R be defined by 

 if  f has a loca l minimum at x =–1, then a possible value of k is

[AIEEE 2010]

Let a, b ∈ R be such that the function f given by   as extreme values at x = –1 and x = 2.

Statement 1 : f has local maximum at x = – 1 and at x = 2
Statement 2: a = 1/2 and = -1/4

[AIEEE 2012]

 [JEE 2000 (Scr.), 1]

Find the area of the right angled triangle of least area that can be drawn so as to circumscribe a rectangle of sides ‘a’ and ‘b’, the right angle of the triangle coinciding with one of the angles of the rectangle.

[REE 2001 Mains, 5]

(a) Let f(x) = (1 + b²) x² + 2bx + 1 and let m(b) is minimum value of f(x).  As b varies, the range of m(b) is

[JEE 2001 (Scr.), 1 + 1]

The maximum value of   under the restrictions   and 

If a₁, a₂ ,......, an are positive real numbers whose product is a fixed number e, the minimum value of   

Find a point on the curve x² + 2y² = 6 whose distance from the line x + y = 7, is minimum.

[JEE 2003 Mains, 2 + 2]

For a circle x² + y² = r² find the value of ‘r’ for which the area enclosed by the tangents drawn from the point P(6, 8) to the circle and the chord of contact is maximum.

Let f(x) = x³ + bx² + cx + d, 0 < b² < c. Then f

[JEE 2004, (Scr.)]

If p(x) be a polynomial of degree 3 satisfying p(–1) = 10, p(1) = –6 and p(x) has maximum at x = –1 and p’(x) has minima at x = 1. Find the distance between the local maximum and local minimum of the curve.

[JEE 2005 Mains, 4]

 If f(x) is cubic polynomial which f(x) has local maximum at x = –1. If f(2) = 18 and f(1) = –1 and f ’(x) has local minima at x = 0, then

[JEE 2006, 5 + 5 + 6]

Let f(x) =  and 

The total number of local maxima and local minima of the function f(x) = 

[JEE 2008, 3 + 4 + 4 + 4]

Comprehension :

Consider the function f :  defined by 

Which of the following is true ?

[JEE 2008]

Comprehension :

Consider the function f :  defined by 

Which of the following is true ?

Which of the following is true ?

Let p(x) be a polynomial of degree 4 having extremum at x = 1, 2 and    Then the value of p(2) is

[JEE 2009]

 The maximum value of the function f(x) = 2x³ – 15x² + 36x – 48 on the set 

 Let f, g and h be real-value d func tions defined on the interval      If a, b and c denote respectively, the absolute maximum of f, g and h on [0, 1], then

The number of distinct real roots of x⁴ – 4x³ + 12x² + x – 1 = 0 is

[JEE 2011, 4]