Question

Let f: R -> R and g: R -> R be differentiable, with g(x) ≠ 0...

Let f: R -> R and g: R -> R be differentiable, with g(x) ≠ 0 for all x. Assume that g(x) f'(x) = f(x) g'(x) for all x. Show that there is a real number c such that f(x) = cg(x) for all x. (Hint: Look at f/g.)

Let g: [0, ∞) -> R, with g(x) = x2 for all x ≥ 0. Let L be the line tangent to the graph of g that passes through the point (0, -6). Find the point (x0, x02) that is the point of tangency.

Homework Answers

Answer #1

if satisfied with the explanation, please rate it up..

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
4a). Let g be continuous at x = 0. Show that f(x) = xg(x) is differentiable...
4a). Let g be continuous at x = 0. Show that f(x) = xg(x) is differentiable at x = 0 and f'(0) = g(0). 4b). Let f : (a,b) to R and p in (a,b). You may assume that f is differentiable on (a,b) and f ' is continuous at p. Show that f'(p) > 0 then there is delta > 0, such that f is strictly increasing on D(p,delta). Conclude that on D(p,delta) the function f has a differentiable...
Consider the function g(x) = |3x + 4|. (a) Is the function differentiable at x =...
Consider the function g(x) = |3x + 4|. (a) Is the function differentiable at x = 10? Find out using ARCs. If it is not differentiable there, you do not have to do anything else. If it is differentiable, write down the equation of the tangent line thru (10, g(10)). (b) Graph the function. Can you spot a point “a” such that the tangent line through (a, f(a)) does not exist? If yes, show using ARCS that g(x) is not...
Let g from R to R is a differentiable function, g(0)=1, g’(x)>=g(x) for all x>0 and...
Let g from R to R is a differentiable function, g(0)=1, g’(x)>=g(x) for all x>0 and g’(x)=<g(x) for all x<0. Proof that g(x)>=exp(x) for all x belong to R.
a) Let f : [a, b] −→ R and g : [a, b] −→ R be...
a) Let f : [a, b] −→ R and g : [a, b] −→ R be differentiable. Then f and g differ by a constant if and only if f ' (x) = g ' (x) for all x ∈ [a, b]. b) For c > 0, prove that the following equation does not have two solutions. x3− 3x + c = 0, 0 < x < 1 c) Let f : [a, b] → R be a differentiable function...
Let f(x) = x^3 - x a) Find the equation of the secant line through (0,f(0))...
Let f(x) = x^3 - x a) Find the equation of the secant line through (0,f(0)) and (2,f(2)) b) State the Mean-Value Theorem and show that there is only one number c in the interval that satisfies the conclusion of the Mean-Value Theorem for the secant line in part a c) Find the equation of the tangent line to the graph of f at point (c,f(c)). d) Graph the secant line in part (a) and the tangent line in part...
Let f : E → R be a differentiable function where E = [a,b] or E...
Let f : E → R be a differentiable function where E = [a,b] or E = (−∞,∞), show that if f′(x) not = 0 for all x ∈ E then f is one-to-one, i.e., there does not exist distinct points x1,x2 ∈ E such that f(x1) = f(x2). Deduce that f(x) = 0 for at most one x.
Let f : R → R be differentiable with derivative f'. Prove that f(x + h)...
Let f : R → R be differentiable with derivative f'. Prove that f(x + h) = f(x) + f'(x)h + o(h), as h → 0.
Let f(x)=22−x2f(x)=22-x2 The slope of the tangent line to the graph of f(x) at the point...
Let f(x)=22−x2f(x)=22-x2 The slope of the tangent line to the graph of f(x) at the point (−4,6) is    . The equation of the tangent line to the graph of f(x) at (-4,6) is y=mx+b for m= and b=   Hint: the slope is given by the derivative at x=−4
Problem: Let y=f(x)be a differentiable function and let P(x0,y0)be a point that is not on the...
Problem: Let y=f(x)be a differentiable function and let P(x0,y0)be a point that is not on the graph of function. Find a point Q on the graph of the function which is at a minimum distance from P. Complete the following steps. Let Q(x,y)be a point on the graph of the function Let D be the square of the distance PQ¯. Find an expression for D, in terms of x. Differentiate D with respect to x and show that f′(x)=−x−x0f(x)−y0 The...
Prove or give a counterexample: If f is continuous on R and differentiable on R∖{0} with...
Prove or give a counterexample: If f is continuous on R and differentiable on R∖{0} with limx→0 f′(x) = L, then f is differentiable on R.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT