Question

Put (v) for true and (x) for false sentences 1. () If f'(a) 0, then f (a) is a local extrema of f. 2. () A function on a closed interval must have a local extrema. 3. ( ) If f have a local max at x = a, then its square f2 must also have a local max at x = a as well 4. () If f and g have both local minima at a, then their sum f +g has a local minima at r-a as well 5. () If f is differentiable and decreasing on (a, b), then f'() < 0 On (a, b

Answer 1

1) True statement: by first derivative test f'(a) = 0 implies 'a' is a loca extremum point.

2) True statement: consider the graph of the function. Draw a parallel line to X-axis. Move the line up and down to get a tangent point to the line. This tangent point will be an extremum.

3) False statement: This need not be true always. For example let f(x) = 1-x in [0,1] , then x = 0 is a local maximum. Now f²(x) = f(f(x)) = f(1-x) = 1-(1-x) = x. But x = 0 is not a local maximum for f²(x) in [0,1].

4) True statement: since 'a' is a local minimum of both f and g we have f'(a) = 0 = g'(a). Now (f+g)'(a) = f'(a)+g'(a) = 0. Thus 'a' is an extremum point of f+g and it will be a local minimum as 'a' is individually local minimum.

5) True statement: since f is decreasing on (a,b) the slope of the curve is negative. Slope of the function curve at x is f'(x) hence f'(x) < 0 on (a,b).