f(x)=−x3 −x2 +16x+16
1. Calculate f′(x).
2. Calculate f′′(x).
3. Find the x values such that f(x) = 0. Note: this can be done by factoring.
4. Find the stationary point/s. Note: a point requires x and y coordinates.
5. Determine the nature of this/these point/s.
6. Find the point/s of inflection.
7. Determine the limits:
limx→∞ f(x) and limx→-∞ f(x).
8. Graph y = f(x) by hand, labelling all important features found in the previous parts. Also, the x and y axes need to be chosen so that all features (including limits) can be seen.
9. Determine the point (x, y) where the function is increasing most rapidly.
What is it you don't understand? We will be happy to critique your work.
To answer the questions, we need to perform some calculations and analysis based on the given function f(x) = -x^3 - x^2 + 16x + 16.
1. To calculate f'(x), we find the derivative of f(x) with respect to x. In this case, we have f'(x) = -3x^2 - 2x + 16.
2. To calculate f''(x), we take the second derivative of f(x). In this case, we have f''(x) = -6x - 2.
3. To find the x values such that f(x) = 0, we can set f(x) equal to zero and solve for x. So, -x^3 - x^2 + 16x + 16 = 0. To factor this equation, you may try factoring by grouping or use numerical methods like the Newton-Raphson method if factoring is not straightforward.
4. To find the stationary points, we need to find the x and y coordinates where f'(x) = 0. In other words, we need to solve the equation -3x^2 - 2x + 16 = 0. Again, this can be done by factoring or using numerical methods.
5. To determine the nature of the stationary points, we can use the second derivative f''(x). If f''(x) > 0, the point is a local minimum. If f''(x) < 0, the point is a local maximum. If f''(x) = 0, further analysis is needed.
6. To find the points of inflection, we need to determine the x values where f''(x) = 0 or where there is a change in the concavity of the function. Again, solve -6x - 2 = 0 to find these points.
7. To determine the limits as x approaches infinity and negative infinity, we can analyze the behavior of the function at these extremes. For lim(x→∞) f(x), determine the highest power of x in the function, and the limit will approach either positive infinity, negative infinity, or some specific value depending on the sign and coefficients of that term. Similarly, for lim(x→-∞) f(x), determine the behavior of the highest power of x.
8. To graph y = f(x) by hand, plot points based on x and f(x) values obtained from previous calculations. Also, label the important features such as x-intercepts, stationary points, points of inflection, etc.
9. To determine the point (x, y) where the function is increasing most rapidly, we can look for the maximum value of f'(x). This can be done by finding the x value(s) where f'(x) attains its maximum. Plug these x values into f(x) to find the corresponding y values. The point(s) with the highest y value will correspond to the function increasing most rapidly.
It is important to note that some of these calculations may involve complex or lengthy processes, requiring a deeper understanding of calculus concepts.