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

Determine the intervals of concavity of f(x)=x4-6x2.

1. Concave down on (-∞,-1) and (1,∞); concave up on (-1,1)
2. Concave up on (-∞,-1) and (1,∞); concave down on (-1,1)
3. Concave up on (-1,1) only
4. Concave up everywhere

Answer

B. Concave up on (-∞,-1) and (1,∞); concave down on (-1,1)

Solution

f''(x)=12x2-12=12(x-1)(x+1). It is positive for |x|>1 (concave up) and negative for -1<x<1 (concave down).

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

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

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