Reading dates for highest/lowest temperatures

I have a spreadsheet recording data from my weather station.
Column headings are Date, Wind (mph), Dewpoint, Outdoor temp ......etc. etc
I have been able to read lowest temperature in a month using LARGE and SMALL (eg =LARGE(J3:J4034,1)), but how can I find the date from the Date column where this highest temperature occurred. And what happens if the highest temperature occurred more than once on two or three different dates?
I have been trying to work out VLOOKUP and HLOOKUP but have not been able to do what I want.
I hope my explanation is clear. In the end I just I want to be able to derive a statement such as the "the lowest temperature in Feb was -7.8ºC on Feb 2nd.

You actually asked the question we'll need answered:

What happens if the highest temperature occurred more than once...?

So...since it's your workbook....What do you WANT to happen?
First occurrence of the highest temp?
Last occurrence of the highest temp?
Something else?

I attached a workbook with a mock-up.
(I hope it works on your Mac)

With
Col_A containing dates...in A2:A50
Col_A containing temps...in B2:B50

Then....
Max Temp
D2: =MAX($B$2:$B$50)

Instance_1 of the Max Temp (regular formula)

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OR...You can use this shorter ARRAY FORMULA, but you must remember to use CTRL+SHIFT+ENTER (instead of just ENTER) when committing it

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Either way, for subsequent instance dates....copy that formula across Row_2

Is that something you can work with?

Re:

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Since we are copying the formula across to the right from E2
we want to know when to stop finding dates and start displaying
text blanks ("").
• This part:

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calculates the count of values that match the MAX TEMP

• This part:

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Counts the columns referenced.
When copied across..it returns:
1 for COLUMNS($E:E)
2 for COLUMNS($E:F)
etc

So this combination determines when the count of MAX TEMP days
has been reached:

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For the "VALUE IF TRUE" calculation...
The ROW NUMBER for each referenced cell is
multiplied by 1 (if the cell equals the MAX TEMP)
or by 0 if it does not.
The result is an array of 0's interspersed with row numbers.

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The SMALL function looks in that array for the nTH smallest value
after the count of 0's (which is the count of cells that DO NOT
equal the MAX TEMP):

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So, if there are 3 MAX TEMP days out of 49 cells,
this combination finds the 47 smallest value
(46 non-matches + the row of the first match):

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That value is the row of the matched value.
The INDEX function that wraps around the first part
of the SMALL function eliminates the need for CTRL+SHIFT+ENTER.

Finally, the outer INDEX function returns the date associated
with the calculated row.

I hope that helps.