1

Problem

Find the values of the constant 'a' that make the function \$f(x)=ax^2 + 3x\$ continuous at x = 2.

2

Step

For a function to be continuous at a particular point, three conditions must be satisfied.

3

Hint

Direct substitution for f(2):

First, let’s find the value of the function at x = 2

\$ f(2) = a(2)^2 + 3(2) = 4a + 6 \$

4

Step

Limit as x approaches 2:
Next, we need to find the limit of the function as x approaches 2

\$\lim_{x \to 2} {(ax^2 + 3x)} \$

5

Step

Since this is a polynomial function, the limit is simply the function evaluated at x = 2

\$\lim_{x \to 2} {(ax^2 + 3x)} = a(2)^2 + 3(2) = 4a + 6\$

6

Step

Continuity condition:
For the function to be continuous at x = 2, the above two expressions must be equal:

\$\lim_{x \to 2} f(x) = f(2) \$
4a + 6 = 4a + 6 (This is always true regardless of the value of a)

7

Solution

Therefore, there are no specific values of 'a' required for the function to be continuous at x = 2. It’s inherently continuous at that point for any value of 'a'.

8

Sumup

Please summarize Problem, Clue, Hint, Formula, Steps and Solution

Choices

9

Choice-A

This option accurately reflects the conclusion we reached. The continuity condition is independent of the specific value of 'a'

Correct Any real number

10

Choice-B

This is incorrect. The value of "a" can be anything

Wrong Positive

11

Choice-C

This implies that 'a' needs to be a specific value (-3) for continuity. However, the limit and function hold true for any 'a', making this option incorrect

Wrong a = - 3

12

Choice-D

This option claims that only when 'a' is 0 will the function be continuous. However, like the previous options, it’s incorrect since 'a' doesn’t affect the continuity at x = 2

Wrong a = 0

13

Answer

Option

A

14

Sumup

Please summarize choices