Math / Calculus

Question3. Find the absolute maximum and absolute minimum of the functions on the given intervals. (a) $\qquad$ $3-6 x^{2}+5$ (d) $f(x)=x+\frac{1}{x}$ , $[1 / 4,4]$ (C)

Studdy Solution

STEP 1

1. We are finding absolute extrema (maximum and minimum values) of given functions on specified intervals.
2. The functions are continuous on their respective intervals.
3. For part (a), the function is $f(x) = 3 - 6x^2 + 5$ .
4. For part (d), the function is $f(x) = x + \frac{1}{x}$ on the interval $[1/4, 4]$ .

STEP 2

1. Identify the critical points of the function within the interval.
2. Evaluate the function at the critical points and endpoints of the interval.
3. Compare these values to determine the absolute maximum and minimum.

STEP 3

For part (a), the function is $f(x) = 3 - 6x^2 + 5$ .
Simplify the function: $f(x) = 8 - 6x^2$ .
Find the derivative: $f'(x) = -12x$ .
Set the derivative equal to zero to find critical points: $-12x = 0$ .
Solve for $x$ : $x = 0$ .

STEP 4

Evaluate $f(x)$ at the critical point and endpoints of the interval.
Endpoints are not specified for part (a), so assume the function is evaluated at $x = 0$ .
Calculate $f(0) = 8 - 6(0)^2 = 8$ .

STEP 5

For part (d), the function is $f(x) = x + \frac{1}{x}$ .
Find the derivative: $f'(x) = 1 - \frac{1}{x^2}$ .
Set the derivative equal to zero to find critical points: $1 - \frac{1}{x^2} = 0$ .
Solve for $x$ : $\frac{1}{x^2} = 1$ implies $x^2 = 1$ , so $x = 1$ .

STEP 6

Evaluate $f(x)$ at the critical points and endpoints of the interval $[1/4, 4]$ .
Calculate $f(1) = 1 + \frac{1}{1} = 2$ .
Calculate $f(1/4) = 1/4 + 4 = 4.25$ .
Calculate $f(4) = 4 + \frac{1}{4} = 4.25$ .

STEP 7

Compare the values obtained:
For part (a), the absolute maximum is $8$ at $x = 0$ .
For part (d), the absolute maximum is $4.25$ at both $x = 1/4$ and $x = 4$ , and the absolute minimum is $2$ at $x = 1$ .
The absolute maximum for part (a) is $8$ at $x = 0$ .
The absolute maximum for part (d) is $4.25$ at $x = 1/4$ and $x = 4$ , and the absolute minimum is $2$ at $x = 1$ .

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Question3. Find the absolute maximum and absolute minimum of the functions on the given intervals. (a) $\qquad$ $3-6 x^{2}+5$ (d) $f(x)=x+\frac{1}{x}$ , $[1 / 4,4]$ (C)

Studdy Solution

STEP 1

STEP 2

1. Identify the critical points of the function within the interval.
2. Evaluate the function at the critical points and endpoints of the interval.
3. Compare these values to determine the absolute maximum and minimum.

STEP 3

For part (a), the function is $f(x) = 3 - 6x^2 + 5$ .
Simplify the function: $f(x) = 8 - 6x^2$ .
Find the derivative: $f'(x) = -12x$ .
Set the derivative equal to zero to find critical points: $-12x = 0$ .
Solve for $x$ : $x = 0$ .

STEP 4

Evaluate $f(x)$ at the critical point and endpoints of the interval.
Endpoints are not specified for part (a), so assume the function is evaluated at $x = 0$ .
Calculate $f(0) = 8 - 6(0)^2 = 8$ .

STEP 5

For part (d), the function is $f(x) = x + \frac{1}{x}$ .
Find the derivative: $f'(x) = 1 - \frac{1}{x^2}$ .
Set the derivative equal to zero to find critical points: $1 - \frac{1}{x^2} = 0$ .
Solve for $x$ : $\frac{1}{x^2} = 1$ implies $x^2 = 1$ , so $x = 1$ .

STEP 6

Evaluate $f(x)$ at the critical points and endpoints of the interval $[1/4, 4]$ .
Calculate $f(1) = 1 + \frac{1}{1} = 2$ .
Calculate $f(1/4) = 1/4 + 4 = 4.25$ .
Calculate $f(4) = 4 + \frac{1}{4} = 4.25$ .

STEP 7

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Question3. Find the absolute maximum and absolute minimum of the functions on the given intervals. (a) \qquad 3−6x2+53-6 x^{2}+53−6x2+5 (d) f(x)=x+1xf(x)=x+\frac{1}{x}f(x)=x+x1​, [1/4,4][1 / 4,4][1/4,4] (C)

Studdy Solution

STEP 1

STEP 2

1. Identify the critical points of the function within the interval.2. Evaluate the function at the critical points and endpoints of the interval.3. Compare these values to determine the absolute maximum and minimum.

STEP 3

STEP 4

Evaluate f(x) f(x) f(x) at the critical point and endpoints of the interval.Endpoints are not specified for part (a), so assume the function is evaluated at x=0 x = 0 x=0.Calculate f(0)=8−6(0)2=8 f(0) = 8 - 6(0)^2 = 8 f(0)=8−6(0)2=8.

STEP 5

STEP 6

STEP 7

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Question3. Find the absolute maximum and absolute minimum of the functions on the given intervals. (a) $\qquad$ $3-6 x^{2}+5$ (d) $f(x)=x+\frac{1}{x}$ , $[1 / 4,4]$ (C)

1. Identify the critical points of the function within the interval.
2. Evaluate the function at the critical points and endpoints of the interval.
3. Compare these values to determine the absolute maximum and minimum.

Evaluate $f(x)$ at the critical point and endpoints of the interval.
Endpoints are not specified for part (a), so assume the function is evaluated at $x = 0$ .
Calculate $f(0) = 8 - 6(0)^2 = 8$ .