91Ó°ÊÓ

Polynomial functions are expressions involving variables raised to whole number exponents and multiplied by coefficients, summed together to form a single expression. For example, a general polynomial function can be represented as:
\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \rightarrow + a_1 x + a_0 \] where \(a_n, a_{n-1}, \ldots, a_0\) are constants, and \(n\) is a non-negative integer.
Understanding polynomial functions is crucial because they allow us to model complex behaviors in many fields like physics, engineering, and economics. In our exercise, we are dealing with the function: \[ f(x) = -2x^4 - 3x + 5 \]
This specific polynomial function is of the fourth degree (the highest power of \(x\) is 4), which means it can have up to three turning points.

The derivative of a polynomial function measures the rate at which the function's value changes as its input changes. Mathematically, it's the function's slope at any given point and is denoted as \(f'(x)\).
To find the derivative of our function, \(f(x) = -2x^4 - 3x + 5\), we differentiate term-by-term: - The derivative of \(-2x^4\) is \(-2 \times 4x^3 = -8x^3\)- The derivative of \(-3x\) is \(-3\)- The derivative of the constant \(5\) is \(0\)Thus, the derivative is: \[ f'(x) = -8x^3 - 3 \]

Critical points of a function are where the derivative equals zero or is undefined. These points often indicate local maxima, minima, or turning points. For our function, we find critical points by setting the derivative equal to zero: \[ -8x^3 - 3 = 0 \] and solving for \(x\). Isolating \(x^3\), we get: \[ x^3 = -\frac{3}{8} \] Taking the cube root gives: \[ x \rightarrow \rightarrow \rightarrow -0.67 \] Verify that this \(x\)-value is within the given interval \( -1 \rightarrow 0 \) and we see that it is part of the valid interval.

Function evaluation involves finding the corresponding \(y\)-coordinate for a given \(x\)-value. Given an \(x\rightarrow \rightarrow -0.67\), we substitute back into the original function to get \(f(x)\). So, we evaluate: \[ f(-0.67) = -2(-0.67)^4 - 3(-0.67) + 5 \] Calculate each term: - \(-2(-0.67)^4 = -2 \times 0.2017 = -0.4034\)- \(-3(-0.67) = 2.01\)- Add \(5\)Combining these, we predict the \(y\)-coordinate: \[ f(-0.67) \rightarrow \rightarrow \rightarrow 6.01 \]

Approximations are useful for finding values that don't have exact solutions or are too cumbersome for manual calculation. They help give meaningful results with a specified degree of accuracy. In our problem, after calculating our critical point \(x \rightarrow \rightarrow -0.67\), the next step was to approximate the exact \(y\)-coordinate. Using a calculator, we concluded that \( f(-0.67) \rightarrow \rightarrow \rightarrow 6.01 \), giving us the turning point's coordinates of roughly \((-0.67, 6.01)\). This result helps understand the nature of the polynomial's graph and predict its behavior accurately within a reasonable margin.