Engineering Board Exam Reviewer · Curve Analysis: Critical Points, Concavity, and Inflection · medium

Find the local maximum value of f(x)=x3-3x2+4.

1. 0
2. 2
3. 8
4. 4

Answer

D. 4

Solution

f'(x)=3x2-6x=3x(x-2)=0 x=0,2. Since f''(x)=6x-6, f''(0)<0 gives a local maximum at x=0: f(0)=4. (At x=2 there is a local minimum, f(2)=0.)

More Curve Analysis: Critical Points, Concavity, and Inflection questions

1. On which interval is f(x)=x2-4x increasing?
2. Find the critical number of f(x)=x2-6x+5.
3. Find the absolute minimum value of f(x)=x4-4x3+10.
4. How many inflection points does f(x)=x5-5x4 have?
5. Find the maximum value of f(x)=xe-x for x≥0.
6. For what value of k does f(x)=x3+kx2 have an inflection point at x=2?
7. Find the absolute maximum value of f(x)=x3-3x on the interval [0,2].
8. Determine the intervals of concavity of f(x)=x4-6x2.
9. Find the x-coordinate of the inflection point of f(x)=x3-6x2+9x+1.

All Curve Analysis: Critical Points, Concavity, and Inflection questions