Find (A) f^'(x) (B) The slope of the graph of f at x=2 and...

Find
(A) $f^{\prime}(x)$
(B) The slope of the graph of $f$ at $x=2$ and $x=4$
(C) The equations of the tangent lines at $x=2$ and $x=4$
(D) The value(s) of $x$ where the tangent line is horizontal
$$
f(x)=3 x^{4}-6 x^{2}-7
$$

Key Concepts

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It gives the instantaneous rate of change and represents the slope of the tangent line at any given point on the function's graph.

Slope of a Function at a Point

The slope of a function at a specific point is determined by the derivative evaluated at that point. This value indicates how steep the graph is at that location and whether the function is increasing or decreasing.

Tangent Line Equation

The tangent line to a function at a certain point is the line that touches the graph of the function at that point without crossing it (locally). Its equation can be derived using the point-slope form, where the slope is the derivative at that point and the point is a coordinate on the function.

Horizontal Tangent

A horizontal tangent line occurs when the derivative of the function is zero. This situation indicates that the slope of the function is flat at that point, which is a key condition when searching for local maxima, minima, or points of inflection.

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