sketch the graph of a differentiable function y = f(x) through the point (-4,4) if f(-4)=0 and satisfies the…

Explanation:

Step1: Recall derivative - function relationship

The sign of the derivative $f^{\prime}(x)$ determines the increasing - decreasing nature of the function $y = f(x)$. If $f^{\prime}(x)>0$, the function is increasing; if $f^{\prime}(x)<0$, the function is decreasing. Also, $f^{\prime}(a) = 0$ indicates a possible local extremum at $x = a$.

Step2: Analyze part (a)

Given $f^{\prime}(-4)=0$, $f^{\prime}(x)>0$ for $x < - 4$ and $f^{\prime}(x)<0$ for $x>-4$. This means the function is increasing on the interval $(-\infty,-4)$ and decreasing on the interval $(-4,\infty)$. So, the function has a local maximum at $x = - 4$. The graph that represents this is a parabola opening downwards with a vertex at $(-4,4)$. So the answer for part (a) is B.

Step3: Analyze part (b)

Given $f^{\prime}(-4)=0$, $f^{\prime}(x)<0$ for $x < - 4$ and $f^{\prime}(x)>0$ for $x>-4$. This means the function is decreasing on the interval $(-\infty,-4)$ and increasing on the interval $(-4,\infty)$. So, the function has a local minimum at $x=-4$. The graph that represents this is a parabola opening upwards with a vertex at $(-4,4)$. So the answer for part (b) is A.

Step4: Analyze part (c)

Given $f^{\prime}(-4)=0$ and $f^{\prime}(x)>0$ for $x\neq - 4$. This means the function is increasing on $(-\infty,-4)\cup(-4,\infty)$. The graph has a horizontal tangent at $x = - 4$ but still increases on either side of $x=-4$. There is no local extremum. The graph is a curve that has a horizontal tangent at $(-4,4)$ and goes up on both sides, which is represented by C.

Step5: Analyze part (d)

Given $f^{\prime}(-4)=0$ and $f^{\prime}(x)<0$ for $x\neq - 4$. This means the function is decreasing on $(-\infty,-4)\cup(-4,\infty)$. The graph has a horizontal tangent at $x = - 4$ but still decreases on either side of $x=-4$. There is no local extremum. The graph is a curve that has a horizontal tangent at $(-4,4)$ and goes down on both sides, which is represented by D.

Answer:

(a) B (b) A (c) C (d) D