(c) h(x) = x|x, 9. Prove that if f: R R is an even function (that is, f(-x)3f(x) for all x E R] and has a derivative at every point, then the derivative f is an odd function [that is, f'(-x) = -f'(x) for all x E R]. Also prove that if g: R R is a differentiable odd function, then g' is an even function. (0)= 0 Show thatg is
(c) h(x) = x|x, 9. Prove that if f: R R is an even function (that is, f(-x)3f(x) for all x E R] and has a derivative at every point, then the derivative f is an odd function [that is, f'(-x) = -f'(x) for all x E R]. Also prove that if g: R R is a differentiable odd function, then g' is an even function. (0)= 0 Show thatg is
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![(c) h(x) = x|x|,
9. Prove that if f: R R is an even function [that is, f(-x)=f(x) for all x E R] and has a
derivative at every point, then the derivative f is an odd function [that is, f'(-x) =-f'(x) for
all x E R]. Also prove that if g : R R is a differentiable odd function, then g is an even
function.
D ha defined by o(x) :=
xsin (1/x2) for x + 0, and g(0) := 0. Show that
is](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb0bc31fe-ee3c-473a-b5eb-76fb2ccf59ad%2F7d7e720c-b306-4c76-b11e-f750adf3e1db%2F72bbwv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(c) h(x) = x|x|,
9. Prove that if f: R R is an even function [that is, f(-x)=f(x) for all x E R] and has a
derivative at every point, then the derivative f is an odd function [that is, f'(-x) =-f'(x) for
all x E R]. Also prove that if g : R R is a differentiable odd function, then g is an even
function.
D ha defined by o(x) :=
xsin (1/x2) for x + 0, and g(0) := 0. Show that
is
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