The sum of the maximum and minimum values of the function f(x) = |5x - 7| + [x² + 2x] in the interva

Q: Why do we check x = 1.4 for the minimum? Shouldn't we just check the endpoints?

That is a fantastic question! In a standard continuous function, we often look at endpoints. However, the modulus function |5x - 7| has a 'corner' or a sharp turn at x = 1.4. Since the modulus is non-negative and hits zero at x = 1.4, it is the absolute lowest point for that specific component. Because the greatest integer part is also well-behaved there, we must test this point to see if it yields the global minimum.

Q: Do I need to check every integer value for the GIF?

Not necessarily. Since x^2 + 2x is strictly increasing on our interval, the greatest integer function will only change values at specific points. You just need to know the range of the quadratic to see which integers it hits. Once you know the range is [4.0625, 8], you know the GIF will only take the values 4, 5, 6, 7, and 8.

Q: Why is the maximum at x = 2?

Think about the behavior of the two components. As x increases from 1.4 to 2, the modulus part |5x - 7| is climbing up the right side of its 'V' shape, and the greatest integer part [x^2 + 2x] is also stepping upwards. When you add two increasing functions together, the resulting sum is also increasing. Therefore, the maximum must occur at the rightmost boundary of our interval, which is x = 2.

Question

The sum of the maximum and minimum values of the function $f(x) = |5x - 7| + [x^2 + 2x]$ in the interval $[\frac{5}{4}, 2]$, where $[t]$ is the greatest integer $\le t$ is ________

Sigma Percentile · Accepted Answer

### Analyzing the Setup

When you see a function like $f(x) = |5x - 7| + [x^2 + 2x]$ on the interval $[\frac{5}{4}, 2]$, your first instinct might be to panic. The secret is to stop looking at the function as a single entity and start seeing it as a team of two distinct players: the modulus function and the greatest integer function.

First, consider the quadratic expression inside the greatest integer bracket: $g(x) = x^2 + 2x$. If we complete the square, we get:

$$g(x) = (x + 1)^2 - 1$$

On our interval $[1.25, 2]$, this function is strictly increasing. It starts at $g(1.25) = 4.0625$ and climbs steadily to $g(2) = 8$. This tells us that the greatest integer part $[x^2 + 2x]$ is a predictable staircase that steps through the integers $4, 5, 6, 7,$ and $8$ as $x$ moves from $1.25$ to $2$.

### Finding the Minimum

Now, let us look at the second player: the modulus function $h(x) = |5x - 7|$. The most critical point for any modulus function is where it hits zero, as this is its absolute minimum. Setting $5x - 7 = 0$ gives us $x = 1.4$.

Crucially, $1.4$ lies right inside our interval $[1.25, 2]$. To find the minimum value of the entire function $f(x)$, we test this critical point:

$$f(1.4) = |5(1.4) - 7| + [1.4^2 + 2(1.4)]$$
$$f(1.4) = 0 + [1.96 + 2.8] = [4.76] = 4$$

Since the modulus cannot be negative and the greatest integer part never drops below $4$ in this interval, **$4$ is our absolute minimum**.

### Finding the Maximum

As we move from $x = 1.4$ toward $x = 2$, notice the behavior of both components. The modulus function is climbing up the right side of its 'V' shape, and the greatest integer function is stepping upward.

Since both components are increasing, their sum must also be increasing. Therefore, the maximum value must occur at the very end of our interval, at $x = 2$:

$$f(2) = |5(2) - 7| + [2^2 + 2(2)]$$
$$f(2) = |10 - 7| + [4 + 4] = 3 + 8 = 11$$

### Final Calculation

We have found our two extremes: the minimum is $4$ and the maximum is $11$. The question asks for the sum of these values:

$$11 + 4 = 15$$

By dividing the problem into its constituent parts and analyzing their behavior, we have tamed the beast. The final answer is **$15$**.

Animated Solution for Mathematics - Functions: The sum of the maximum and minimum values of the function $f (x) = ∣5 x - 7∣ + [x^{2} + 2 x]$ in the interval $[\frac{5}{4}, 2]$ , where $[t]$ is the greatest integer $\leq t$ is ________

Enter Numerical Value:

Visualized Solution

Understanding the Function

Analyzing the Quadratic Part

Range of the Quadratic Part

Behavior of the Greatest Integer Function

Analyzing the Modulus Part

Finding the Minimum Value

Calculating the Minimum Value

Finding the Maximum Value

Calculating the Maximum Value

Final Summation

The Sigma Insight: Classification of Functions

Analyzing the Setup

Finding the Minimum

Finding the Maximum

Final Calculation

Similar Questions

The absolute minimum value of the function $f (x) = ∣ x^{2} - x + 1∣ + [x^{2} - x + 1]$ , where $[t]$ denotes the greatest integer function, in the interval $[- 1, 2]$ , is :

If $S = {a \in R : ∣2 a - 1∣ = 3 [a] + 2 {a}}$ , where $[t]$ denotes the greatest integer less than or equal to $t$ and ${t}$ represents the fractional part of $t$ , then $72 \sum_{a \in S} a$ is equal to ______

Let $f : R \to R$ be a function defined by $f (x) = (2 + 3 a) x^{2} + (\frac{a + 2}{a - 1}) x + b, a \neq = 1$ If $f (x + y) = f (x) + f (y) + 1 - \frac{2}{7} x y$ , then the value of $28 \sum_{i = 1}^{5} ∣ f (i) ∣$ is

Let $[t]$ denote the greatest integer $\leq t$ . Then the equation in $x$ , $[x]^{2} + 2 [x + 2] - 7 = 0$ has :

If $f (x) = cos [π^{2}] x + cos [- π^{2}] x$ , where $[x]$ stands for the greatest integer function, then

Let $f (x) = 2 x^{n} + λ, λ \in R, n \in N$ , and $f (4) = 133, f (5) = 255$ . Then the sum of all the positive integer divisors of $(f (3) - f (2))$ is

Let $f (x) = [x]^{2} - [x + 3] - 3, x \in R$ , where $[\cdot]$ is the greatest integer function. Then

For $x \in R$ , let $[x]$ denote the greatest integer $\leq x$ , then the sum of the series $[\frac{1}{3}] + [\frac{1}{3} + \frac{1}{100}] + [\frac{1}{3} + \frac{2}{100}] + \dots + [\frac{1}{3} + \frac{99}{100}]$ is

A function $f (x)$ is given by $f (x) = \frac{5 ^{x}}{5 ^{x} + 5}$ , then the sum of the series $f (\frac{1}{20}) + f (\frac{2}{20}) + f (\frac{3}{20}) + \dots + f (\frac{39}{20})$ is equal to:

If a function $f$ satisfies $f (m + n) = f (m) + f (n)$ for all $m, n \in N$ and $f (1) = 1$ , then the largest natural number $λ$ such that $\sum_{k = 1}^{2022} f (λ + k) \leq (2022)^{2}$ is equal to _________