Question
I.Let f(x)= ) 1+4x-x^2&forxleqslant 3 (x+5)/(2)&forxgt 3 (i) Is f a continuous function on the set of real numbers R? (ii) Is f a differentiable function on R? (iii) Find the maximum value and minimum values of f(x) for xin [0,4] (iv) Plot the graph of f(x) on [-4,4]
Solution
4.2
(187 Votos)
Rodrigo
Elite · Tutor por 8 anos
Resposta
(i) To determine if the function
is continuous on the set of real numbers
, we need to check if the function is continuous at every point in
.The function
is defined as a piecewise function with two different expressions for different intervals of
. To check the continuity, we need to ensure that the two expressions are equal at the boundary point
.For
, the function is defined as
.For
, the function is defined as
.To check the continuity at
, we need to evaluate the left-hand limit and the right-hand limit as
approaches 3.Left-hand limit:
Right-hand limit:
Since the left-hand limit and the right-hand limit are equal, the function
is continuous at
.Therefore, the function
is continuous on the set of real numbers
.(ii) To determine if the function
is differentiable on
, we need to check if the function is differentiable at every point in
.The function
is defined as a piecewise function with two different expressions for different intervals of
. To check the differentiability, we need to ensure that the two expressions have the same derivative at the boundary point
.For
, the function is defined as
.For
, the function is defined as
.To check the differentiability at
, we need to evaluate the left-hand derivative and the right-hand derivative as
approaches 3.Left-hand derivative:
Right-hand derivative:
Since the left-hand derivative and the right-hand derivative are not equal, the function
is not differentiable at
.Therefore, the function
is not differentiable on
.(iii) To find the maximum and minimum values of
for
, we need to evaluate the function at the critical points and endpoints of the interval.The critical points occur where the derivative of the function is zero or undefined. In this case, the derivative is undefined at
, so we need to evaluate the function at
.For
, the function is defined as
.For
, the function is defined as
.Evaluating the function at the critical point
:
Now, we need to evaluate the function at the endpoints
and
.
Therefore, the maximum value of
for
is
, and the minimum value is
.(iv) To plot the graph of
on
, we need to evaluate the function for a range of values within the interval
.For
, the function is defined as
.For
f(x