are also differentiable on [a, b] and [c, d],. respectively. Show that (g = f di

Question

are also differentiable on [a, b] and [c, d],. respectively. Show that (g = f differentiable and find the derivative. Suppose f R rightarrow R is differentiable and define g(x) = .x2f . Show that g is and compute g'. Define f(x) = for x 0. Determine where f is the derivative. Define f : [0. 2] rightarrow R by f(x) = 2x - x2 Show that f satisfies the . theorem and find r such that (c) = 0. Define f : R rightarrow R by f(x) = 1/(1+ x2). Prove that f has a maximum value and point at which that maximum occurs. Prove that the equation has at most one root in the Show that has exactly one root in Supper f [0, 2] rightarrow R is differentiable, f(0) = 0. F(1) = 2, and f (2) = 2. Prove that then: is such that

Explanation / Answer

f is continuous everywhere, and equals 0 at x=- as well as x=+. So by Rolle's theorem, df/dx must be 0 somewhere in (-,+). That proves f(x) must have an extremum somewhere.

df/dx=-2x/(1+x2)2=0 when x=0, so the extremum occurs when x=0. To prove it is a maximum,

d2f/dx2=(6x2-2)/(1+x2)3=-2<0 when x=0, proving that x=0 is indeed a maximum

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