Question 8.4.1: Find the maximum and minimum values, for x ∈ [0, 3], of: f (......

Essential Mathematics for Economic Analysis [1185377]

Find the maximum and minimum values, for x ∈ [0, 3], of:
$f (x) = 3x² - 6x + 5$

Step-by-Step

The 'Blue Check Mark' means that this solution was answered by an expert.
Learn more on how do we answer questions.

The function is differentiable everywhere, and $f^{\prime}(x) = 6x − 6 = 6(x − 1).$ Hence x = 1 is the only critical point. The candidate extreme points are the end points 0 and 3, as well as x = 1. We calculate the value of $f$ at these three points. The results are $f (0) = 5, f (3) = 14, and f (1) = 2.$ We conclude that the maximum value is 14, obtained at x = 3, and the minimum value is 2 at x = 1.

Related Answered Questions

Question: 8.3.3

A monopolist is faced with the inverse demand function P(Q) denoting the price when output is Q. The monopolist has a constant average cost k per unit produced. (a) Find the profit function π(Q), and prove that the first-order condition for maximal profit at Q^∗> 0 is P(Q^∗) + Q^∗P'(Q^∗) = k (∗) (b) ...

Verified Answer:

(a) The profit function is π(Q) = QP(Q) − kQ, so [...

Question: 8.1.1

Find possible maximum and minimum points for: (a) f (x) = 3 − (x − 2)² (b) g(x) =√x − 5 − 100, for x ≥ 5 ...

Verified Answer:

(a) Because

(x − 2)^{2} ≥ 0

for all...

Question: 8.3.2

Suppose that the total cost of producing Q > 0 units of a commodity is C(Q) =aQ² + bQ + c, where a, b, and c are positive constants. (a) Find the value of Q that minimizes the average cost defined by A(Q) = C(Q)/Q in the special case when C(Q) = 2Q² + 10Q + 32. (b) Show that in the general case, ...

Verified Answer:

(a) We find that here A(Q) = 2Q + 10 + 32/Q, so [l...

Question: 8.3.1

Suppose Y(N) bushels of wheat are harvested per acre of land when N pounds of fertilizer per acre are used. If P is the dollar price per bushel of wheat and q is the dollar price per pound of fertilizer, then profits in dollars per acre are π(N) = PY(N) − qN for N ≥ 0. Suppose there exists N^∗ such ...

Verified Answer:

(a) The profit function is

\pi(N)=P Y(N)-q ...

Question: 8.7.2

Find possible inflection points for f (x) = 1/9x³ − 1/6x² − 2/3x + 1. ...

Verified Answer:

From Example 8.6.1, we have

f^{\prime\prime...

Question: 8.7.1

Show that the function f (x) = x^4 does not have an inflection point at x = 0, even though f”(0) = 0. ...

Verified Answer:

Here

f^{\prime }(x) = 4x^{3}

and [l...

Question: 8.7.3

Find possible inflection points for f (x) = x^6 − 10x^4. ...

Verified Answer:

In this case

f^{\prime}(x)=6x^{5}-40x^{3}{\...

Question: 8.4.2

Find the maximum and minimum values, for x ∈ [−1, 3], of f (x) = 1/4x^4 − 5/6x³ + 1/2x² − 1 ...

Verified Answer:

The function is differentiable everywhere, and [la...

Question: 8.4.3

Test the mean value theorem on the function f (x) = x³ − x, defined over [0, 2]. ...

Verified Answer:

We find that [f (2) − f (0)]/(2 − 0) = 3 and [late...

Question: 8.2.1

Measured in milligrams per litre, the concentration of a drug in the bloodstream, t hours after injection, is given by the formula c(t) = t/t² + 4, t ≥ 0 Find the time of maximum concentration. ...

Verified Answer:

Differentiating with respect to t yields

c^...