Question

I cannot answer this question involving derivatives and graph sketching, hope somebody can help me solve it!
Thanks in advance

Answer 1

The derivative of a function, found by using the power rule, is a graph with roots (x-intercepts) that are the same as the x-coordinates of the turning points of the said function;
This means that since the graph f(x) in the question shows us that the x-coordinates of the two turning points (A and B) are -2 and 6, we can determine that:
f'(x) = n(x + 2)(x - 6)
because if this was set to 0, the solutions would be x = -2 and x = 6;
The variable n represents the constant that could very well be 1 but we cannot assume so because of the limited information given;
This variable n would not change the solutions, but it will different for two functions that are the same in almost every respect (i.e. they will have the same x-coordinates of their turning points and roots) except that one is more vertically stretched than the other.

In any case, all you need to worry about for this question is drawing a u-shaped curve that crosses the x-axis at -2 and 6;
Don't worry about where it crosses the y-axis or how low it dips, just make all these things moderate as they cannot be deduced from the information given;
Its lowest point will be at x = 2 as the curve should be symmetric so just make sure the bottom of the curve is around that point.

Hope this helps.

Answer 2

Graphically, the derivative of a function is the slope of the tangent line for each point. In the first figure, some points have a segment of their tangent line drawn. Between x = -6 and x = -2, the slopes are positive and decrescent as x reach -2. In point x = -2 the slope is equal to 0. Between x = -2 and x = 2, the slopes are negative and decrescent as x reach 2. Between x = 2 and x = 6, slopes still are negative but with crescent values, until x = 6 where slope is zero. Between x = 6 and x = 10, slopes are positive and crescent. In the second figure, f'(x) is shown.