Chapter 3: Q. 2 (page 286)

Critical points:
Find the critical points of each of the following
functions.
$f (x) = \frac{3 x - 2}{x - 1} f (x) = \frac{1}{1 + \sqrt{x}} f (x) = \sin (\frac{π}{2} x) f (x) = e^{x} (x - 2)$

Short Answer

Expert verified

The critical point of the function $f (x) = \frac{3 x - 2}{x - 1}$ is $x = 1$ .

The critical point of the function $f (x) = \frac{1}{1 + \sqrt{x}}$ is $x = 0, 1$ .

The critical point of the function $f (x) = \sin (\frac{π}{2} x)$ is $x = 1, 3, 5, . . .$ .

The critical point of the function $f (x) = e^{x} (x - 2)$ is $x = 1$ .

Step by step solution

Step 1. Given Information.

The functions are:

$f (x) = \frac{3 x - 2}{x - 1} f (x) = \frac{1}{1 + \sqrt{x}} f (x) = \sin (\frac{π}{2} x) f (x) = e^{x} (x - 2)$

Step 2. Critical points.

The critical points of the function $y = f (x)$ are values c such that $f' (c) = 0 o r c$ value does not exist.

Step 3. Consider the function.

Consider the function, $f (x) = \frac{3 x - 2}{x - 1}$

$f' (x) = \frac{(x - 1) (3) - 3 x - 2 (1)}{{(x - 1)}^{2}} = - \frac{1}{{(x - 1)}^{2}}$

$f' (x)$ does not occur at $x = 1$ .

Step 4. Consider the function.

Consider the function, $f (x) = \frac{1}{1 + \sqrt{x}}$

$f' (x) = - \frac{1}{{(1 + \sqrt{x})}^{2}} \frac{1}{2 \sqrt{x}} {(1 + \sqrt{x})}^{2} = 1 \sqrt{x} = - 1 x = 1 A n d \sqrt{x} = 0 x = 0$

Critical points are $x = 0, 1$

Step 5. Determine the critical points.

Consider the function $f (x) = \sin (\frac{π}{2} x)$ ,

$f' (x) = \frac{π}{2} \cos (\frac{π}{2} x) = 0 \cos (\frac{π}{2} x) = 0 \frac{π}{2} x = (2 n + 1) \frac{π}{2}, n = 1, 2, 3, . . .$

The derivative is zero when $x = 1, 3, 5, . . .$

Step 6. Determine the critical points.

Consider the function,

$f (x) = e^{x} (x - 2) f' (x) = e^{x} (x - 1) = 0 x - 1 = 0 x = 1$

The critical point is $x = 1$ .

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Critical points:
Find the critical points of each of the following
functions.
$f (x) = \frac{3 x - 2}{x - 1} f (x) = \frac{1}{1 + \sqrt{x}} f (x) = \sin (\frac{π}{2} x) f (x) = e^{x} (x - 2)$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Critical points.

Step 3. Consider the function.

Step 4. Consider the function.

Step 5. Determine the critical points.

Step 6. Determine the critical points.

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Critical points:Find the critical points of each of the followingfunctions.f(x)=3x-2x-1f(x)=11+xf(x)=sin(π2x)f(x)=ex(x-2)

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Critical points.

Step 3. Consider the function.

Step 4. Consider the function.

Step 5. Determine the critical points.

Step 6. Determine the critical points.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Discrete Mathematics

Geometry

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Critical points:
Find the critical points of each of the following
functions.
$f (x) = \frac{3 x - 2}{x - 1} f (x) = \frac{1}{1 + \sqrt{x}} f (x) = \sin (\frac{π}{2} x) f (x) = e^{x} (x - 2)$