Question

sketch a graph of the function ( f ) that is continuous on ( (-infty,infty) ) and has the following properties ( f^{prime}(x)>0, f^{prime prime}(x)>0 ) choose the correct graph below. ( circ a ). ( circ b ). ( circ c ). ( circ d ).

Explanation:

Step1: Analyze the first - derivative condition

Since \(f^{\prime}(x)>0\), the function \(f(x)\) is increasing on \((-\infty,\infty)\). A function \(y = f(x)\) is increasing when, as \(x\) increases, \(y\) also increases.

Step2: Analyze the second - derivative condition

Since \(f^{\prime\prime}(x)>0\), the function \(f(x)\) is concave up on \((-\infty,\infty)\). A concave - up function has the property that the slope of the tangent line is increasing.

Answer:

B.