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Chapter 1: Problem 60

Use a graphing utility to graph the function and approximate (to two decimal places) any relative minimum or relative maximum values. $$ f(x)=-2 x^{2}+9 x $$

Short Answer

Expert verified
The approximate relative maximum of the function \(f(x)=-2x^2+9x\) is 5.06.

Step by step solution

01

Identify the coefficients

In the quadratic function \(f(x)=-2x^2+9x\), the coefficient \(a\) is -2, and \(b\) is 9.
02

Determine the x-coordinate of the vertex

Calculate \(h\), which represents the x-coordinate of the vertex. This can be determined by the formula: \(h=-b/(2a)\). So, \(h=-9/(2*(-2))= 2.25\).
03

Determine the y-coordinate of the vertex

Calculate \(k\), which represents the y-coordinate of the vertex. Substitute the value of \(h\) into \(f(x)=-2x^2+9x\) to get \(k\) . So, \(k=f(2.25)= -2*(2.25)^2+9*(2.25)= 5.0625\).
04

Graph the function

Plot the vertex (2.25 , 5.0625) and graph the downward-opening parabola. The location of the vertex can be confirmed by seeing that it is the maximum point of the parabola.
05

Approximate the relative minimum or maximum value

In this case, since the parabola opens downwards, there will be a relative maximum value which is located at the vertex. So the approximate relative maximum value to two decimal places is 5.06.

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