AP DSC School Assistant Math Model Test 1

Concept:The nature of a function at a critical point (where derivative changes sign) can be determined using the first derivative test. If the derivative changes from positive to negative as we pass through the point, the function has a local maximum; if from negative to positive, it is a local minimum.Explanation:Given $f(x) = \begin{cases} x^{3}+x^{2}+10x, & x<0 \\ -3\sin x, & x\ge 0 \end{cases}$.We examine the derivative from left and right of $x = 0$.For $x < 0$:$f'(x) = 3x^{2}+2x+10$.At $x = 0$ (approaching from left), $f'(0) = 10 > 0$. So the function is increasing just before 0.For $x \ge 0$:$f'(x) = -3\cos x$.At $x = 0$, $f'(0) = -3 < 0$. So the function is decreasing just after 0.Since the derivative changes from positive to negative at $x = 0$, the function has a local maximum at $x = 0$.Answer:Option A. There is a point of maximum.