In this question, if I have to verify that pi/3 is the point of maxima via first derivative test then - Maths - Application of Derivatives

Question

In this question, if I have to verify that pi/3 is the point
of maxima via first derivative test then why does it give this as
the point of minima???
Prove that pi/3 is the point of maxima via first derivative
test???

Q 25 If the sum of lengths of the hypotenuse and a side of a right
angles triangle is given, then show that the area of triangle is
maximum, where the angle between them is π 3 ,

Sandeep Saurav · Accepted Answer

Dear Student, As per your query we are proving π3 is the point of maxima via first derivative test So, We are providing you solution after putting dAdθ=0⇒k22(1+cos θ)3(2 cos2θ+cos θ-1)=0⇒2 cos2θ+cos θ-1=0⇒2 cos2θ+2cos θ-cos θ-1=0⇒2 cos θ(cos θ+1)-1(cos θ+1)=0⇒(cos θ+1)(2cos θ-1)=0either cos θ=-1 or cos θ=12either θ=π or θ=π3First Derivative TestHere we shall have three cases case 1 : θ<π3Take a value in this interval and check whether dAdθ is positive or negativefor convenient take θ=0 ⇒dAdθθ=0=k22(1+cos 0)3(2 cos20+cos 0-1)=k22(1+1)3(2×1+1-1)=k22(2)3×2=k28>0⇒Positivecase 2 : π3<θ<πTake a value in this interval and check whether dAdθ is positive or negativefor convenient take θ=π2 ⇒dAdθθ=π2=k22(1+cos π2)3(2 cos2π2+cos π2-1)=k22(1+0)3(2×0+0-1)=k22(-1)=-k22<0⇒Negativecase 1 : θ>πTake a value in this interval and check whether dAdθ is positive or negativefor convenient take θ=2π ⇒dAdθθ=2π=k22(1+cos 2π)3(2 cos22π+cos 2π-1)=k22(1+1)3(2×1+1-1)=k22(2)3×2=k28>0⇒PositiveSo, at π3 slope (dAdθ) is changing from Positive to negative⇒Maxima at π3 So, at π slope (dAdθ) is changing from Negative to Positive⇒Minima at π Hope this information will clear your doubts about topic If you have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible Keep posting!! Regards