Over which interval is the graph of [tex]f(x)=-x^2+3x+8[/tex] increasing?

A. [tex](-\infty, 1.5)[/tex]
B. [tex](-\infty, 10.25)[/tex]
C. [tex](1.5, \infty)[/tex]
D. [tex](10.25, \infty)[/tex]



Answer :

To determine the interval over which the graph of the function [tex]\( f(x) = -x^2 + 3x + 8 \)[/tex] is increasing, follow these steps:

1. Find the derivative of the function [tex]\( f(x) \)[/tex]:
The first step is to differentiate the function [tex]\( f(x) \)[/tex] to find its derivative [tex]\( f'(x) \)[/tex]. The derivative of the function gives us the slope of the tangent line at any point [tex]\( x \)[/tex] on the graph. Calculating the derivative,

[tex]\[ f'(x) = \frac{d}{dx}(-x^2 + 3x + 8) = -2x + 3 \][/tex]

2. Determine where the derivative is positive:
A function is increasing where its derivative is positive. So, we need to find the values of [tex]\( x \)[/tex] for which [tex]\( f'(x) > 0 \)[/tex].

Set the inequality:

[tex]\[ -2x + 3 > 0 \][/tex]

Solve for [tex]\( x \)[/tex]:

[tex]\[ -2x + 3 > 0 \][/tex]
[tex]\[ 3 > 2x \][/tex]
[tex]\[ \frac{3}{2} > x \][/tex]
[tex]\[ x < 1.5 \][/tex]

Thus, [tex]\( f(x) \)[/tex] is increasing for [tex]\( x < 1.5 \)[/tex].

3. Write the interval where the function is increasing:
Based on the solution to the inequality, [tex]\( f(x) \)[/tex] is increasing on the interval [tex]\( (-\infty, 1.5) \)[/tex].

Therefore, the interval over which the graph of [tex]\( f(x) = -x^2 + 3x + 8 \)[/tex] is increasing is:

[tex]\[ \boxed{(-\infty, 1.5)} \][/tex]