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
$$
f(x)=x^{4}-32 x^{2}+10
$$

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Key Concepts

Differentiation of Polynomial Functions

This concept involves using the power rule, which states that the derivative of x^n is n*x^(n-1), to find the rate at which a polynomial function changes with respect to its variable. It is fundamental in calculus for understanding behavior such as increasing or decreasing trends and finding slopes at specific points.

Slope of a Tangent Line

The slope of a tangent line at a given point on a graph represents the instantaneous rate of change of the function at that point. It is computed by evaluating the derivative of the function at that specific x-value, providing a precise measure of how steeply the function is moving at that moment.

Equation of a Tangent Line

Once the slope of the tangent line at a point is known, the equation of the tangent line can be constructed using the point-slope form of a line, which is y - f(a) = f'(a)(x - a). This gives a linear approximation of the function near the point of tangency.

Horizontal Tangency

The concept of a horizontal tangent involves finding points where the tangent line is horizontal, meaning the slope is zero. This requires setting the derivative of the function equal to zero and solving for the corresponding x-values, which often indicates local extrema or points of inflection on the graph.

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