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How many distinct solutions does the following equation have? log(base x^2 +2) (4-5x^2 - 6x^3) = 2 a)None, b)1, c)2, d)4, e)Infinitely many

log(base x 2 +2) (4-5x 2 - 6x 3 ) = 2 is equivalent to 4-5x 2 -6x 3 =(x 2 +2) 2 Therefore, we are looking for solutions to the equation 4 - 5x 2 - 6x 3 = x 4 + 4x 2 + 4 => x 4 + 6x 3 + 9x 2 = 0 => x 2 ...
TM
Answered by Teodora M. • MAT tutor
2345 Views

Let a and b be positive real numbers. If x^2 + y^2<=1 then what is the largest that ax+by can get?

By the Cauchy-Schwartz inequality, we have (x 2 + y 2 )(a 2 +b 2 ) &gt;= (ax+by) 2 . This can be transformed into (ax+by) 2 &lt;= (x 2 + y 2 )(a 2 +b 2 ) &lt;= 1 * (a 2 +b 2 ) &lt;= (a 2 +b 2 ). Hence ax + b...
TD
Answered by Tutor135762 D. • MAT tutor
4410 Views

Given a + b = 20, find the maximum value of ba^2.

In the question we are given that a + b = 20 where a and b are positive integers. Remember this.First let y = a^2 b. This is a good starting point as we have an equality now which we can mess with.We want to...
SB
Answered by Samuel B. • MAT tutor
3932 Views

We define the digit sum of a non-negative integer to be the sum of its digits. For example, the digit sum of 123 is 1 + 2 + 3 = 6. Let n be a positive integer with n < 10. How many positive integers less than 1000 have digit sum equal to n?

First we should consider how many positive integers less than 100 have digit sum equal to n. The answer should be relatively straight forward for this - it is n+1.For example, if n = 3 then we can have 03,12...
RL
Answered by Ron L. • MAT tutor
6508 Views

How do you differentiate ln(f(x))? Tricks like these occur commonly in STEP questions (including one I was looking at earlier today).

Assumed knowledge: The chain rule, implicit differentiation and being able to differentiate exp(x).Chain rule: d/dx = d/dy * dy/dxImplicit differentiation: f(y) = g(x) =&gt; f'(y)*dy/dx = g'(x) (where f'(y) ...
JR
Answered by Joseph R. • MAT tutor
2698 Views