Let be a real-valued function that is continuous on and differentiable on
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
Let f be a real-valued function that is continuous on [a,b] and differentiable on (a,b). Suppose that c∈(a,b) satisfies f′(c)=0 and that for every x in (a,b),
f′′(x)≥m>0,
where m is a positive constant.
(a) Prove that f has a strict local minimum at x=c. Your argument must begin with the definition of a local extremum and proceed directly from the given inequality.
(b) Using only the hypotheses stated above and the result of part (a), prove that f(c)≤f(x) for every x in [a,b]. Structure your reasoning on interval placement of x relative to c and avoid invoking any extremum theorems beyond those derivable from first principles.