consider the function y = f(x) graphed below. give the x - coordinate of a point where: a. the derivative of…
consider the function y = f(x) graphed below. give the x - coordinate of a point where: a. the derivative of the function is positive: x = b. the value of the function is positive: x = c. the derivative of the function is largest: x = d. the derivative of the function is zero: x = e. the derivative of the function is approximately the same as the derivative at x = 0.75 (be sure that you give a point that is distinct from x = 0.75!): x =
Answer
Explanation:
Step1: Recall derivative - function relationship
The derivative of a function is positive when the function is increasing. Looking at the graph, we can see the function is increasing around (x = 3).
Step2: Determine where function value is positive
The function value (y = f(x)) is positive when the graph is above the (x) - axis. For example, (x = 2) has (y>0).
Step3: Find where derivative is largest
The derivative is largest where the slope of the tangent line to the graph is steepest and positive. This occurs around (x = 3) (where the function is increasing most rapidly).
Step4: Locate where derivative is zero
The derivative of a function is zero at local maxima, minima or horizontal points. Here, at the local minimum and local maximum, the derivative is zero. For example, at the local - minimum point around (x = 0.5) the derivative is zero.
Step5: Find point with similar derivative as at (x = 0.75)
We need to find a point with a similar slope of the tangent line as at (x = 0.75). Looking at the symmetry of the graph, a point around (x = 1.25) might have a similar slope.
Answer:
A. (x = 3) B. (x = 2) C. (x = 3) D. (x = 0.5) E. (x = 1.25)