Suppose that u=g(x) is differentiable at x=1 and that y=f(u)...

Suppose that $u=g(x)$ is differentiable at $x=1$ and that $y=f(u)$ is differentiable at $u=g(1) .$ If the graph of $y=f(g(x))$ has a horizontal tangent at $x=1,$ can we conclude anything about the tangent to the graph of $g$ at $x=1$ or the tangent to the graph of $f$ at $u=g(1) ?$ Give reasons for your answer.

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Key Concepts

Zero-Product Implications

When the product of two numbers is zero, as in f'(g(1))·g'(1) = 0, it implies that at least one of the factors must be zero. However, without additional information, it is impossible to determine for certain which specific derivative is zero. This concept is central in understanding that a horizontal tangent in the composite function does not guarantee that either the tangent to the graph of the inner function g or the outer function f must be horizontal individually.

Chain Rule

The chain rule is a fundamental concept in differential calculus that applies to compositions of functions. It states that the derivative of a composite function f(g(x)) is the product f'(g(x))·g'(x). This rule is crucial when analyzing how changes in the inner function g(x) and the outer function f(u) impact the overall rate of change of the composite function.

Horizontal Tangents

A horizontal tangent at a point on a graph indicates that the derivative of the function at that point is zero. This means that the rate of change of the function is zero at that specific point, often corresponding to a local maximum, minimum, or saddle point. In the context of composite functions, a horizontal tangent implies that the product of the derivatives involved equals zero.

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