Question

$f(x)=\int\limits_{\cos x}^{\sin x}\left(1-t+2 t^3\right) d t$ 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)

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