Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
AP EAMCET201920 Apr 2019Morning ShiftMathematicsApplication of DerivativesActual

For (m > 1, n > 1 ), the value of (c ) for which the Rolle's theorem is applicable for the function (f(x)=x^ 2 m-1 (a-x)^ 2 n ) in ((0, a) ) is

Options

  1. A( 2 a m-1 m+2 n-1 )
  2. B( a(m-n+1) 2 m+2 n )
  3. C( a(2 m-1) 2 m+2 n-1 )
  4. D( a(2 m+1) m+n-1 )

Correct answer

C. ( a(2 m-1) 2 m+2 n-1 )

Step-by-step solution

Since it is given that Rolle's theorem is applicaple for the function (f(x)=x^ 2 m-1 (a-x)^ 2 n ) in ((0, a). ) ( aligned & So, & f^ (x)=(2 m-1) x^ 2 m-2 (a-x)^ 2 n -2 n(a-x)^ 2 n-1 x^ 2 m-1 & at x=c, m > 1, n > 1 & f^ (c)=0 & (2 m-1) c^ 2 m-2 =2 n c^ 2 m-1 (a-c)^ 2 n-1 & (2 m-1) c = 2 n a-c & c= a(2 m-1) 2 n+2 m-1 aligned ) Hence, option (3) is correct.

Practice Application of Derivatives on Quantrex Academy →

More from Application of Derivatives

Consider the quadratic equation a x^2+b x+c=0 , where 2 a+3 b+6 c=0 and let g(x)= a x^3 3 + b x^2 2 +c x . Statement-I : The given quadratic equation ax ^2+ bx + c =0 has at least 2025The difference between the absolute maximum and absolute minimum values of the function f(x)=2 x^3-15 x^2+36 x-30 on [-1,4] is 2025If f(x)=x e^ x(1-x) , x R , then f(x) is 2025The angle between the curves y ^2= x and x ^2= y at the point (1,1) is 2025If the tangent of the curve 4 y^3=3 a x^2+x^3 drawn at the point (a, a) forms a triangle of area 25 24 sq.units with the coordinate axes then a = 2025If the function f(x)= x- ^2 x is defined on the interval [- , ] , then f is strictly increasing in the interval 2025If the Lagrange's mean value theorem is applied to the function f(x)=e^x defined on the interval [1,2] and the value of c (1,2) is k , then e^ k-1 = 2025If the tangent to the curve x y+a x+b y=0 at (1,1) makes an angle Tan ⁻¹ 2 with X -axis, then ab a + b = 2025 Full Application of Derivatives list All AP EAMCET PYQs