Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Roots of Equation.
The number of roots of the equation \(x^2+sin^2{x}-1\) in the closed interval \([0,\frac{\pi}{2}]\) is
Equation
Roots
Algebra
Answer:2
B.Stat Objective Problem 711
Challenges and Thrills of Pre-College Mathematics by University Press
\(x^2+sin^2{x}-1=0\)
\(\Rightarrow x^{2}=cos^{2}x\)
we draw two graphs \(y=x^{2} and y=cos^{2}x\)
where intersecting point gives solution now we look for intersecting points
we get two intersecting points
so number of roots is 2.
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Another wrong solution! The proof is wrong as it only shows that there is at least one solution. It does not show that there is exactly one solution. To complete the proof, show that the derivative of \(f(x)\) is non-negative in the interval \([0, \pi/2]\).