5. The height (feet) of a projectile x seconds after it is launched straight up in the air is given by f(x) =-16x? + 214x +8. Find its maximum height.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
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### Problem Statement
The height (feet) of a projectile \(x\) seconds after it is launched straight up in the air is given by the function \(f(x) = -16x^2 + 214x + 8\). Find its maximum height.
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### Explanation
This problem involves finding the maximum height reached by a projectile when its height as a function of time is given by a quadratic equation \(f(x) = -16x^2 + 214x + 8\).
In a quadratic equation in standard form, \(f(x) = ax^2 + bx + c\), where \(a < 0\) indicates the parabola opens downward, the maximum height can be found at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula:
\[ x = \frac{-b}{2a} \]
For the given function \(f(x) = -16x^2 + 214x + 8\):
- \(a = -16\)
- \(b = 214\)
Substitute \(a\) and \(b\) into the vertex formula to find the time \(x\) at which the maximum height occurs.
\[ x = \frac{-214}{2 \cdot -16} = \frac{-214}{-32} = 6.6875 \]
Now, substitute \(x = 6.6875\) back into the original function to find the maximum height.
\[ f(6.6875) = -16(6.6875)^2 + 214(6.6875) + 8 \]
Perform the calculation to get the maximum height.
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### Detailed Calculation
1. Find the x-coordinate of the vertex:
\[ x = \frac{-214}{2 \cdot -16} = \frac{-214}{-32} = 6.6875 \]
2. Substitute \( x = 6.6875 \) back into the function to get the height:
\[ f(6.6875) = -16(6.6875)^2 + 214(6.6875) + 8 \]
\[ f(6.6875) = -16(44.736) + 214(6.6875) + 8 \]
\[ f(6.6875) = -715.776](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7e54067-4748-4999-a1d1-1be0cd123028%2F54e205e5-a855-4acc-9666-4a43ce4847bd%2F4stx8ug_processed.jpeg&w=3840&q=75)
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