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2 answers:
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Given:
The given function is:
To find:
The range of the given function.
Solution:
We have,
It is a quadratic function because the highest power of the variable x is 2.
Here, the leading coefficient is -32 which is negative. So, the graph of the given function is a downward parabola.
If a quadratic function is , then the vertex of the quadratic function is:
In the given function, .
The value of the given function at is:
The vertex of the given downward parabola is . It means the maximum value of the function is
. So,
Therefore, the range of the given function is .
The series converges to 1/(1-9x) for -1/9<x<1/9
Given the series is ∑
We have to find the values of x for which the series converges.
We know,
∑ converges to (a) / (1-r) if r < 1
Otherwise the series will diverge.
Here, ∑ is a geometric series with |r| = | 9x |
And it converges for |9x| < 1
Hence, the given series gets converge for -1/9<x<1/9
And geometric series converges to a/(1-r)
Here, a = 1 and r = 9x
Therefore, a/(1-r) = 1/(1-9x)
Hence, the given series converges to 1/1-9x for -1/9<x<1/9
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