AP EAMCET202125 Aug 2021Evening ShiftMathematicsApplication of DerivativesActual
The points on the curve y= 2 3 x^3+ 1 2 x^2 , where the tangents make equal angles with coordinate axes are
Options
- A( 1 2 , 5 24 ) and (-1, -1 6 )
- B(1, 7 6 ) and (-1, -1 6 )
- C(2, 22 3 ) and ( 1 2 , 5 24 )
- D(1, 7 6 ) and (2, 22 3 )
Correct answer
A. ( 1 2 , 5 24 ) and (-1, -1 6 )
Step-by-step solution
y= 2 3 x^3+ 1 2 x^2 aligned d y d x & = 2 3 (3 x^2 )+ 1 2 (2 x) & =2 x^2+x aligned Tangents make equal angles with coordinate axis. Hence, d y d x =1 aligned & 2 x^2+x=1 & 2 x^2+2 x-x-1=0 & 2 x(x+1)-1(x+1)=0 & (x+1)(2 x-1)=0 x= 1 2 ,-1 & aligned At aligned x & = 1 2 , y= 2 3 ( 1 2 )^3+ 1 2 ( 1 2 )^2 & = 1 12 + 1 8 = 5 24 y & = 5 24 aligned At x=-1, y= 2 3 (-1)^3+ 1 2 (-1)^2= -2 3 + 1 2 gathered y= -1 6 ( 1 2 , 5 24 ) and (-1, -1 6 ) gathered