Subjects calculus

Turning Points Explanation 139Bf9

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Turning Points Explanation 139Bf9


1. Let's first understand what a turning point is in the context of a function. 2. A turning point occurs where the function changes direction from increasing to decreasing or vice versa. Mathematically, this happens where the first derivative $f'(x)$ is zero and the second derivative $f''(x)$ indicates a change in concavity. 3. If a function has no turning points, it means its derivative $f'(x)$ does not have any real roots where the slope changes sign. 4. For example, consider the function $f(x) = e^x$. Its derivative is $f'(x) = e^x$, which is always positive and never zero. 5. Since $f'(x)$ never equals zero, the function is always increasing and has no turning points. 6. Therefore, the absence of turning points means the function is either strictly increasing or strictly decreasing throughout its domain without any local maxima or minima.