91Ó°ÊÓ

A derivative is a fundamental concept in calculus, representing the rate at which a function changes at any given point. It essentially tells us how the function's output value will change in response to a change in the input value.
\[ f'(x) \] is used to denote the derivative of a function \( f(x) \). To compute it, you differentiate the function, following rules like the power rule, product rule, chain rule, etc. For the function \( f(x) = 3 - 4x - x^2 \), applying the power and linearity principles gives us \( f'(x) = -4 - 2x \).
This derivative function provides valuable insights, indicating whether the function is increasing or decreasing. If \( f'(x) > 0 \), the function is increasing, and if \( f'(x) < 0 \), it is decreasing. When \( f'(x) = 0 \), we have potential critical points.

A tangent line to a curve at a given point is a straight line that just "touches" the curve at that point. It indicates the instantaneous rate of change of the function at that specific point.
The equation of a tangent line at a point \( x = a \) is given by:

• Point: \( (a, f(a)) \)
• Slope: \( f'(a) \)
• Equation: \( y - f(a) = f'(a)(x - a) \)

For the function \( f(x) = 3 - 4x - x^2 \), if we find the derivative as \( f'(x) = -4 - 2x \), this tells us the slope of the tangent line at any point \( x \).
But for a tangent line to be horizontal, the slope, or derivative, must be zero.

A horizontal tangent line is a tangent line with a slope of zero, essentially a flat line that does not rise or fall. Finding where this occurs helps identify specific behaviors of a function.
For the function \( f(x)=3-4x-x^2 \), to find when the tangent line is horizontal, we set its derivative \( f'(x) = -4 - 2x \) equal to zero:

• \(-4 - 2x = 0\)
• Solve for \( x \): \( x = 2 \)

Once the derivative is zero, it confirms that the tangent line at \( x = 2 \) is indeed horizontal. This information can help in detecting flat maximums or minimums on the graph of the function.

Critical points of a function are locations where the derivative is zero or fails to exist. These points are useful for identifying areas where the function may reach local maxima, minima, or points of inflection.
For the function \( f(x) = 3 - 4x - x^2 \), the critical point is found by setting the derivative equal to zero:

• Derivative: \( f'(x) = -4 - 2x \)
• Set equal to zero: \( -4 - 2x = 0 \)
• Solve for \( x \): \( x = 2 \)

At \( x = 2 \), it is essential to check the behavior of the function by substituting back into the original function \( f(x) \), which gives \( f(2) = -5 \). Thus, \((2, -5)\) is a critical point, indicating that the graph has a horizontal tangent line here, possibly a local maximum or minimum.