Chapter 6: Q36. (page 291)

Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of the function.
$f (x) = 4 x - x^{2} + 1$

Short Answer

Expert verified

The function has maximum value that is 5.

Step by step solution

Step 1. Use the concept.

Consider the function $f (x) = a x^{2} + b x + c, a \neq 0$ , the x-coordinate of vertex is $\frac{- b}{2 a}$ .

The graph of $f (x) = a x^{2} + b x + c, a \neq 0$

opens up and has a minimum value when $a > 0$ , and
opens down and has a maximum value when $a < 0$

Step 2. Given Information.

The given function is $f (x) = 4 x - x^{2} + 1 = - x^{2} + 4 x + 1$

Step 3. Solution.

In the function $f (x) = - x^{2} + 4 x + 1$ , we have $a = - 1, b = 4, c = 1$

Here, $a = - 1 < 0$

So, the graph opens down and has a maximum value.

The maximum value of the function is the y-coordinate of the vertex.

Thex-coordinate of the vertex is

$\begin{array}{l} \frac{- b}{2 a} \\ = \frac{- (4)}{2 (- 1)} .......... [a = - 1, b = 4] \\ = 2 \end{array}$

Find they-coordinate of the vertex by evaluating the function for $x = 2$ .

$f (2) = - {(2)}^{2} + 4 (2) + 1 = 5$

So, the maximum value of function is 5.

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Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of the function.
$f (x) = 4 x - x^{2} + 1$

Short Answer

Step by step solution

Step 1. Use the concept.

Step 2. Given Information.

Step 3. Solution.

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Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of the function.f(x)=4x-x2+1

Short Answer

Step by step solution

Step 1. Use the concept.

Step 2. Given Information.

Step 3. Solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of the function.
$f (x) = 4 x - x^{2} + 1$