$f(x)=x^3+8x^3+2x-15$
Answer (2)
The function f ( x ) = 9 x 3 + 2 x − 15 has one real root at approximately x ≈ 1.13 , is always increasing, and has an inflection point at x = 0 .
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Simplify the function: f ( x ) = 9 x 3 + 2 x − 15 .
Find the real root: x \tapprox 1.13 .
Determine that the function is always increasing since 0"> f ′ ( x ) = 27 x 2 + 2 > 0 .
Find the inflection point at x = 0 since f ′′ ( x ) = 54 x changes sign at x = 0 .
The function has a root at x ≈ 1.13 , is always increasing, and has an inflection point at x = 0 .
Explanation
Simplify the function First, we need to simplify the given function. The function is given as f ( x ) = x 3 + 8 x 3 + 2 x − 15 . Combining like terms, we get f ( x ) = 9 x 3 + 2 x − 15 .
Find the roots Next, we want to find the roots of the function. This means we want to solve the equation 9 x 3 + 2 x − 15 = 0 . This is a cubic equation, which can be difficult to solve analytically. However, we can use numerical methods or tools to approximate the roots. Using a numerical method, we find that there is one real root approximately at x \tapprox 1.13 .
Find the first derivative To analyze the behavior of the function, we can find its first derivative. The first derivative is f ′ ( x ) = 27 x 2 + 2 . Since x 2 is always non-negative, 27 x 2 + 2 is always positive. This means that the function is always increasing.
Find the second derivative Now, let's find the second derivative. The second derivative is f ′′ ( x ) = 54 x . Setting f ′′ ( x ) = 0 , we get x = 0 . This is a potential inflection point. When x < 0 , f ′′ ( x ) < 0 , so the function is concave down. When 0"> x > 0 , 0"> f ′′ ( x ) > 0 , so the function is concave up. Therefore, x = 0 is an inflection point.
Summary In summary, the function f ( x ) = 9 x 3 + 2 x − 15 has one real root at approximately x \tapprox 1.13 . The function is always increasing. There is an inflection point at x = 0 .
Examples
Understanding the behavior of cubic functions like this one is useful in many fields, such as physics and engineering, where polynomial models are used to approximate real-world phenomena. For example, the function could represent the volume of a container as a function of one of its dimensions. Finding the roots and inflection points can help determine optimal dimensions or critical points in the system.
