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Chapter 2: Problem 91

Find the values of \(b\) such that the function has the given maximum or minimum value. \(f(x)=-x^{2}+b x-75 ;\) Maximum value: 25

Short Answer

Expert verified
The possible values for \(b\) are \(b = -20\) or \(b = 20\).

Step by step solution

01

Identify the Form Of The Given Function

The given function \(f(x)=-x^{2}+b x-75\) can be written in the standard form of a quadratic function \(ax^{2}+bx+c\), where \(a=-1\), \(b=b\), \(c=-75\). The maximum or minimum value of this function occurs at \(x=-b/(2a)\).
02

Determine the Value of x That Gives the Maximum

Function \(f(x)=-x^{2}+b x-75\) has a maximum value of 25. Let鈥檚 equate \(x\) value in the function to find a formula for \(b\). So substituting \(f(x) = 25\) into the equation we get: \(25=-x^2+bx-75\). Rewriting in standard equation format gives: \(x^{2}-bx+100 = 0\).
03

Find the Value of \(b\)

Now we need to remember that the roots of a quadratic equation \(ax^{2}+bx+c = 0\) sum up to \(-b/a\) and multiply to \(c/a\). In our case, the two roots of the equation we found in the previous step are equal (since we know there is a maximum point), so if we call them \(r\), we get \(r+r=-b/1\), and \(r*r=100/1\). Solving these two equations gives \(r=-b/2\), and \(r^2=100\). Solving the first equation for \(b\) we get \(b=-2r\). Substituting \(r=\sqrt{100}\) into this equation we find the values of \(b\) which are \(b=-2*(-10)\) or \(b=-2*10\).

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