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Chapter 5: Problem 10

Let \(f^{\prime}(x)>0\) and \(f^{\prime \prime}(x)<0\), where \(x_{1}\frac{f\left(x_{1}\right)+2 f\left(x_{2}\right)}{3}$ (b) $f\left(\frac{x_{1}+2 x_{2}}{3}\right)<\frac{f\left(x_{1}\right)+2 f\left(x_{2}\right)}{3}$ (c) $f\left(\frac{x_{1}+2 x_{2}}{3}\right)=\frac{f\left(x_{1}\right)+2 f\left(x_{2}\right)}{3}$ (d) None of these

Short Answer

Expert verified
Since f'(x) > 0, the function is strictly increasing, and with f''(x) < 0, the function is concave down. The graph of the function will lie below the tangent line at x = x鈧�. This leads to the conclusion that \( f\left(\frac{x_{1}+2 x_{2}}{3}\right) > \frac{f\left(x_{1}\right)+2 f\left(x_{2}\right)}{3} \). Therefore, the correct answer is (a).

Step by step solution

01

We are given that f'(x) > 0 and f''(x) < 0 for all x. A positive first derivative indicates that the function is strictly increasing, meaning that f(x鈧�) < f(x鈧�) since x鈧� < x鈧�. A negative second derivative indicates that the function is concave down, meaning that the graph of the function is "curved downward" and lies below its tangent lines. #Step 2: Find the tangent line to the graph of f(x) at x鈧�#:

Since f'(x) > 0, the function is increasing and differentiable, which means that there exists a tangent line at x = x鈧�. Let's find the equation of the tangent line at this point: 1. Calculate the slope of the tangent line: m = f'(x鈧�). 2. Apply the point-slope form of the linear equation using the point (x鈧�, f(x鈧�)): y = f(x鈧�) + f'(x鈧�)(x - x鈧�). #Step 3: Compare averaged values#:
02

Next, we need to evaluate the averaged expressions: 1. Evaluate f((x鈧� + 2x鈧�)/3). 2. Evaluate (f(x鈧�) + 2f(x鈧�))/3. #Step 4: Use the concavity property#:

Since f''(x) < 0, the function is concave down. This means that the graph will lie below the tangent line at x = x鈧�. Since x鈧� > x鈧� and f((x鈧� + 2x鈧�)/3) is on the tangent line at x = x鈧�, while (f(x鈧�) + 2f(x鈧�))/3 is on the graph, we can conclude that f((x鈧� + 2x鈧�)/3) > (f(x鈧�) + 2f(x鈧�))/3. Therefore, the answer is option (a). Solution:
03

Determine monotonicity and concavity

f'(x) > 0 => function is strictly increasing f''(x) < 0 => function is concave down
04

Find the tangent line to the graph of f(x) at x鈧�

m = f'(x鈧�) y = f(x鈧�) + f'(x鈧�)(x - x鈧�)
05

Compare averaged values

f((x鈧� + 2x鈧�)/3) (f(x鈧�) + 2f(x鈧�))/3
06

Use the concavity property

Since function is concave down, f((x鈧� + 2x鈧�)/3) > (f(x鈧�) + 2f(x鈧�))/3. Answer: (a)

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