1.Assume that both the functions f and g are twice differentiable and the second derivatives are never 0 on an interval I. If f and g are concave upward on I, show that f + g is concave upward on I.
1.Assume that both the functions f and g are twice differentiable and
the second derivatives are never 0 on an interval I. If f and g are
concave upward on I, show that f + g is concave upward on I.
2. If f and g are +ve, increasing, concave upward functions on an interval I, show that the product function fg is concave upward on I.
3. Suppose f and g are both concave upward on (−∞, ∞) . Under what
condition on f will the composite function h(x) = f(g(x)) be concave
upward?
4. What are the largest and the smallest possible values taken on by the
product of three distinct numbers spaced so the central number x has
distance 1 from each of the other two, if x lies in (−2, 2)? [Hints: f(x) =
(x-1)x(x+1). Why?]
5. What is the largest value of the product of two numbers x and y, if x and
y are related by x + 2y = a, where a is a fixed positive constant?
6. Let f(x) = x^3 + 3x + 1 = 0. Show that by Intermediate value theorem,
the function f has a real solution and deduce by applying Rolle’s theorem
that indeed f has exactly one solution.
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