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Let [math]f[/math] have a continuous second derivative at every point of an interval containing the number [math]a[/math] in its interior, and let [math]f^\prime(a) = 0[/math]. Show that [math]f[/math] has a local maximum value at [math]a[/math] if [math]f^{\prime\prime} (a) \lt 0[/math], and a local minimum value at [math]a[/math] if [math]f^{\prime\prime} (a) \gt 0[/math]. [Hint: Use the Taylor Formula [math]f(x) = T_1(x) + R_1[/math] and the fact that, if a continuous function is positive (or negative) at [math]a[/math], then it is positive (or negative) near [math]a[/math].]