Question

question 3 (1 point)
if ( f^{prime}(x)>0 ) for all ( x ), then
a) ( fleft(x_{1}
ight)>fleft(x_{2}
ight) ) for every ( x_{1}<x_{2} )
b) ( fleft(x_{1}
ight)<fleft(x_{2}
ight) ) for every ( x_{1}>x_{2} )
c) ( fleft(x_{1}
ight)=fleft(x_{2}
ight) ) for every ( x_{1}<x_{2} )
d) ( fleft(x_{1}
ight)>fleft(x_{2}
ight) ) for every ( x_{1}>x_{2} )

Explanation:

Step1: Recall the relationship between derivative and function monotonicity

If \(f^{\prime}(x)>0\) for all \(x\) in an interval, then the function \(y = f(x)\) is strictly increasing on that interval.
A function \(y = f(x)\) is strictly increasing if for any \(x_1,x_2\) in the domain of \(f\), \(x_1>x_2\) implies \(f(x_1)>f(x_2)\)

Step2: Analyze each option

• Option a: If \(x_1 < x_2\), for a strictly increasing function \(f(x_1)<f(x_2)\), so option a is wrong.
• Option b: If \(x_1>x_2\), for a strictly increasing function \(f(x_1)>f(x_2)\), so option b is wrong.
• Option c: If \(x_1 < x_2\), for a non - constant function (since \(f^{\prime}(x)>0\), the function is non - constant), \(f(x_1)

eq f(x_2)\), so option c is wrong.

• Option d: If \(x_1>x_2\), for a strictly increasing function \(f(x_1)>f(x_2)\) (because when the derivative \(f^{\prime}(x)>0\), the function is increasing).

Answer:

d) \(f(x_1)>f(x_2)\) for every \(x_1 > x_2\)