Let a function f(x) be continuous in an inter | vedclass

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(C) Given that $f(x)$ is continuous in $[a, b]$ and $f(c - \delta)  0$, it follows by the defini

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$f(x)$ has only one local maximum at $c$

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(C) Given that $f(x)$ is continuous in $[a, b]$ and $f(c - \delta)  0$, it follows by the definition of a local maximum that $f(x)$ has a local maximum at $x = c$.Now, consider the condition $(f(\alpha - \delta) - f(\alpha))(f(\alpha + \delta) - f(\alpha)) This inequality implies that the two factors $(f(\alpha - \delta) - f(\alpha))$ and $(f(\alpha + \delta) - f(\alpha))$ must have opposite signs.Case $1$: $f(\alpha - \delta) - f(\alpha) > 0$ and $f(\alpha + \delta) - f(\alpha)  f(\alpha)$ and $f(\alpha + \delta) Case $2$: $f(\alpha - \delta) - f(\alpha)  0$. This implies $f(\alpha - \delta)  f(\alpha)$. This indicates that the function is increasing in the neighborhood of $\alpha$, so $\alpha$ is also a point of inflection.Therefore, $f(x)$ has only one local maximum at $c$ and no local extrema at $\alpha$.

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