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PCAT Quantitative Reasoning – Function Minima and Maxima

  • by Sam
  • Apr 19, 2018
  • PCAT Question of the Day

Which maximum or minimum y value does the function f(x) = 3x2 – 2x + 8 contain?

A. A minimum at y = 3
B. A minimum at y = 29
C. A maximum at y = 3
D. A maximum at y = 29

Click for Explanation

B is correct. To find extrema for a function, we can solve for the first derivative and set the first derivative equal to 0. Essentially this finds minima and maxima where the tangent line is horizontal, or has a slope of 0:

f(x) = 3x2 – 2x + 8
f’(x) = 6x – 2
0 = 6x – 2
2 = 6x
x = 3

So we know that some minimum or maximum occurs at x = 3, but we need to find the y-value:

f(3) = y = 3(3)2 – 2(3) + 8 = 27 – 6 + 8 = 29

Now, we don’t know quite yet whether this is a minimum or maximum value. To do this, we should find the second derivative. If its value is positive, this point represents a minimum, and if the value is negative, this point represents a maximum:

f’(x) = 6x – 2
f’’(x) = 6

Since the second derivative equals a positive value, the point at (3, 29) is a minimum.

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