How do you find the inflection points of the graph of the function: #f(x)= (x^7/42)  (3x^6/10) + (6x^5/5)  (4x^4/3)#?
See the explanation.
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To find the inflection points of the function f(x) = (x^7/42)  (3x^6/10) + (6x^5/5)  (4x^4/3), follow these steps:
 Compute the second derivative of the function f(x) with respect to x.
 Set the second derivative equal to zero and solve for x to find the critical points.
 Test each critical point by substituting it into the second derivative.
 If the second derivative changes sign at a critical point, then it is an inflection point.
Let's proceed with the calculations:

Compute the second derivative of f(x): f'(x) = (7x^6/42)  (18x^5/10) + (30x^4/5)  (16x^3/3) f''(x) = (42x^5/42)  (90x^4/10) + (120x^3/5)  (48x^2/3) = x^5  9x^4 + 24x^3  16x^2

Set the second derivative equal to zero and solve for x: x^5  9x^4 + 24x^3  16x^2 = 0 Factor out x^2: x^2(x^3  9x^2 + 24x  16) = 0 The solutions for x^2 = 0 are x = 0.

Now, we solve for the cubic equation x^3  9x^2 + 24x  16 = 0. One of the solutions is x = 2.

Test each critical point: For x = 0: f''(0) = 0^5  9(0)^4 + 24(0)^3  16(0)^2 = 0. The test is inconclusive. For x = 2: f''(2) = 2^5  9(2)^4 + 24(2)^3  16(2)^2 = 32  9(16) + 24(8)  16(4) = 32  144 + 192  64 = 16. The second derivative is positive.
Therefore, the inflection point is at x = 2.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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