Chapter 6: Q43. (page 291)

Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of the function.
$f (x) = \frac{3}{4} x^{2} - 5 x - 2$

Short Answer

Expert verified

The function has minimum value that is -10.333.

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 $f (x) = \frac{3}{4} x^{2} - 5 x - 2$

Step 3. Solution.

In the function $f (x) = \frac{3}{4} x^{2} - 5 x - 2$ , we have $a = \frac{3}{4}, b = - 5, c = - 2$

Here, $a = \frac{3}{4} > 0$

So, the graph opens up and has a minimum value.

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

Thex-coordinate of the vertex is

$\begin{array}{l} \frac{- b}{2 a} \\ = \frac{- (- 5)}{2 (\frac{3}{4})} .......... [a = \frac{3}{4}, b = - 5] \\ = 3.333 \end{array}$

Find they-coordinate of the vertex by evaluating the function for $x = 3.333$ .

$f (3.333) = \frac{3}{4} {(3.333)}^{2} - 5 (3.333) - 2 = - 10.333$

So, the minimum value of function is -10.333.

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Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of the function.
$f (x) = \frac{3}{4} x^{2} - 5 x - 2$

Short Answer

Step by step solution

Step 1. Use the concept.

Step 2. Given Information.

Step 3. Solution.

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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

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Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of the function.
$f (x) = \frac{3}{4} x^{2} - 5 x - 2$