Application of Derivatives questions
910 published questions with worked solutions.
- For a polynomial g(x) with real coefficients, let m_g denote the number of distinct real roots of g(x). Suppose S is the set of...
- Let the function f: (0, ) R be defined by f( ) = ( + )^2 + ( - )^4. Suppose the function f has a local minimum at precisely when...
- Let f: R R be defined by f(x) = x^2 - 3x - 6 x^2 + 2x + 4 Then which of the following statements is (are) TRUE ?
- Let f: [- 2 , 2 ] R be a continuous function such that f(0) = 1 and _0^ 3 f(t) ,dt = 0 Then which of the following statements is...
- Let |M| denote the determinant of a square matrix M. Let g: [0, 2 ] R be the function defined by g( ) = f( ) - 1 + f ( 2 - ) - 1...
- Let P be a point on the parabola y^2 = 4ax, where a > 0. The normal to the parabola at P meets the x-axis at a point Q. The area...
- Let S be the set of all twice differentiable functions f from R to R such that d^2f dx^2 (x) > 0 for all x (-1, 1). For f S, let...
- For x R , let ^ -1 (x) (- 2 , 2 ). Then the minimum value of the function f: R R defined by f(x) = _0^ x ^ -1 x e^ (t - t) 1 + t^...
- A common tangent T to the curves C_ 1 : x^ 2 /4 + y^ 2 /9 = 1 and C_ 2 : x^ 2 /42 - y^ 2 /143 = 1 does not pass through the...
- Let M and N be the number of points on the curve y^ 5 - 9xy + 2x = 0, where the tangents to the curve are parallel to x-axis and...
- If m and n respectively are the number of local maximum and local minimum points of the function f(x) = _ 0 ^ x^ 2 (t^ 2 - 5t +...
- The number of distinct real roots of x^ 4 - 4x + 1 = 0 is :
- The function f(x) = xe^ x(1−x) , x R, is
- The sum of the absolute maximum and absolute minimum values of the function f(x) = ^ -1 ( x − cos x) in the interval [0, ] is
- If the minimum value of f(x) = 5x^ 2 /2 + /x^ 5 , x > 0 is 14, then the value of is equal to -
- The minimum value of the twice differentiable function f(x) = _ 0 ^ x e^ x-t f'(t)dt - (x^ 2 - x + 1)e^ x , x R, is
- The number of real solutions of x^ 7 + 5x^ 3 + 3x + 1 = 0 is equal to _____.
- Let l be a line which is normal to the curve y = 2x^ 2 + x + 2 at a point P on the curve. If the point Q(6, 4) lies on the line l...
- If the tangent to the curve y = x^ 3 - x^ 2 + x at the point (a, b) is also tangent to the curve y = 5x^ 2 + 2x - 25 at the point...
- Let f(x) = 3^ (x^ 2 - 2)^ 3 + 4 , x R. Then which of the following statements are true ? P : x = 0 is a point of local minima of...
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