Question

$f(x)=\underset{\cos x}{\overset{\sin x}{\int }}\left(1-t+2t^3\right)dt$ has in $[0,2\pi ]$

• A. a maximum at $\frac{\pi }{4}\&$ a minimum at $\frac{3\pi }{4}$
• B. a maximum at $\frac{3\pi }{4}$ & a minimum at $\frac{7\pi }{4}$
• C. a maximum at $\frac{5\pi }{4}$ \& a minimum at $\frac{7\pi }{4}$
• D. neither a maxima nor minima

Answer 1

Answer: (B)

$f^{\prime }(x)=\left(1\cdot \cos x-\sin x\cos x+2{\sin }^{3}x\cdot \cos x\right)-\left(-1\cdot \sin x+\cos x\sin x-2{\cos }^{3}x\sin x\right)$
$=\cos x+\sin x-2\sin x\cdot \cos x+2\sin x\cdot \cos x=\cos x+\sin x$