Let . The following diagram shows the graph of . The -intercept is at and intersects -axis at C and…
Mathematics20256 marksNumerical
Let $f(x) = \cos x + \sqrt{3}\sin x, 0 \le x \le 2\pi$.
The following diagram shows the graph of $f$.
The $y$-intercept is at $(0, 1)$ and intersects $x$-axis at C and D. There is a minimum point at $A(p, q)$ and a maximum point at B.
Based on the above information, answer the questions that follow.
(a)[1.0]
Write $f'(x)$ in the form of $\lambda\cos(x+\mu)$.
(b)[1.0]
Find the value of $q$.
(c)[1.0]
Find the coordinate of the point B.
(d)[1.0]
Find the interval $f(x)$ is decreasing.
(e)[1.0]
Find the slope of the tangent to the curve at D.
(f)[1.0]
Find the slope of the normal to the curve at C.
Answer
Answer (a)
Official answer key
$2\cos\left(x + \frac{\pi}{6}\right)$
Final answer: $2\cos\left(x + \frac{\pi}{6}\right)$
Answer (b)
Official answer key
$-2$
Final answer: -2
Answer (c)
Official answer key
$\left(\frac{\pi}{3}, 2\right)$
Final answer: $\left(\frac{\pi}{3}, 2\right)$
Answer (d)
Official answer key
$\left(\frac{\pi}{3}, \frac{4\pi}{3}\right)$
Final answer: $\left(\frac{\pi}{3}, \frac{4\pi}{3}\right)$
